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# QUADRATIC EQUATION BY FACTORING METHOD

If all methods to solve quadratic equations are included in your syllabus, then these words “solve quadratic equations by factoring method” are not new to you. But it’s not necessary that every student is efficient in solving factorization. So, if you are also escaping from factorization and you are finding the factoring quadratics calculator then this explanation is for you. Before starting the explanation of the factoring method to solve quadratic equations we know about the factorization.

## FACTORIZATION

In mathematics any number which is exactly divisible by the other number then the other number will be the factor of that number. For instance, 12 is exactly divisible by 1, 2, 3, 4, 6, & 12 then all these numbers are the factors of 12. So, any expression which is expressed in a form of product of two or more factors is known as factorization.

The second-degree polynomial equation with its standard form as ax2 + bx + c = 0 is known as Quadratic equation and the formula for Quadratic equation is as:

(x = \dfrac{ -b \pm \sqrt{b^2 – 4ac}}{ 2a })

To solve quadratic equations is easy and by using the online Quadratic formula calculator this task becomes very easy with great accuracy of the result. But when the quadratic equations are solved by the factoring method then it is necessary to get know how about the process of factoring to become the best Quadratic equation solver. Let’s understand the process of factoring to solve quadratic equations.

## FACTORING PROCESS TO SOLVE QUADRATIC EQUATIONS

In the process of the factoring of the form of ax2 + bx + c = 0, we consider that the factors a and b of c (which is a third term) are found such that their sum (a + b) is equal to b (which is the coefficient of the middle term). For finding the factors of ax2 + bx + c = 0, we follow the following steps:

• To find “a” the coefficient of x2, “b” the coefficient of x and “c” the constant term.
• To find two numbers b & c such that b + c = b and bc = ac.
• The factors of ax2 + bx + c will be (ax + b) and (x + c).

### EXAMPLE #1:

Solve the equation 2a2 + a – 1 by factoring method.

### SOLUTION:

Here a = 2, b = 1 & c = -1.

First, we multiply a with c = 2 × -1 = -2, now find the factors of -2 which are 1 & 2 and when we take it as 2 – 1 = 1 and 1 = a so, we may write it as 2a – a = a which is equal to b.

2a2 + 2a – a – 1

2a (a + 1) – 1(a + 1)

(2a -1) (a +1).

### EXAMPLE #2:

Solve the equation 6a2 + 11a – 10.

### SOLUTION:

Here a = 6, b = 11, and c = -10

By the multiplication of a and c we get 6×-10 = -60, now find the factors of -60.

1× -60, 2 × -30, 3 × -20, 5 × -12, 6 × -10, 10 × -6, 12 × -5, 15 × -4 & 60 × -1.

From the above factors of -60 if we take

15 – 4 = 11 and when we multiply these numbers, we will get the product of a × c = 60. So, 15a – 4a = 11a.

6a2 + 15a – 4a – 10.

3a (2a + 5) – 2(2a + 5).

(3a – 2) (2a + 5).

From the above discussion it will be clear to you how you will solve quadratic equations by factoring method. But if you have less time to submit your assignment and you need to find the quadratic function calculator then try our factoring quadratics calculator to solve quadratic equations in minutes with accurate results.

# Quadratic Equation Is Based On Squared

“Formula for Quadratic equation” these words come to your mind when you start to solve Quadratic equations as a new topic in your study. As we know quad means double that’s why one variable in the Quadratic equation is based on squared. Basically, the word quad is a Latin word and when we solve the quadratic equation, we find it in its standard form as $$ax^2 + bx + c=0$$ The most important method to solve quadratic equations is the Quadratic formula $$x = \dfrac{ -b \pm \sqrt{b^2 – 4ac}}{ 2a }$$ The solution of the quadratic equation by this formula is very easy. We just put the values of a, b, & c in the formula then solve the equation according to the structure of the quadratic formula and get required solution sets. In this modern & scientific era online Quadratic formula calculator is also available to solve the quadratic equations in minutes with the accuracy of the result. Now we discuss who invented the Quadratic formula? or the history of the Quadratic equation formula.

# History Of The Formula For Quadratic Equation

## Babylonians Contributions:

Through the Babylonian’s clay tablets as early as 2000 BC the Babylonian mathematicians used quadratic equations by geometric method to find out the area related problems of rectangles. They solved the quadratic equation only for the positive roots and the process which they used to find the values of x and y was as under:

• First, they calculated the half value of p.

• Secondly, they got the square of the result.

• In the third step they subtracted the value of q.

• In the fourth step they found the positive square root of the values.

• Finally, they added the result of the first and 4th step to get the value of x.

The above-mentioned steps may be written as in the form of formula:

$$x = \dfrac{p}{2} + \sqrt{(\dfrac{p}{2})^2-q}$$

The Babylonian mathematicians didn’t use the geometrical method as in the form of formula but if we compare the above-mentioned formula with the today formula as $$x=\dfrac{-b+\sqrt{b^2-4ac}}{2a}$$ then in the today formula $$a=1, b=-p, \& \space c = q \space$$ So, we can say that it became the origin of the formula for the quadratic equation.

solve the quadratic equation by the quadratic formula $$x^2 – 6x + 8 = 0$$

Solution:

Here $$a=1,b=-6, \space \& \space c=8$$

$$x = \dfrac{-b\pm\sqrt{b^2-4ac}}{ 2}$$

$$x=\dfrac{-(-6)\pm\sqrt{(-6)^2-4(4)(8)}}{2(1)}$$

$$x=\dfrac{6\pm\sqrt{36-32 }}{ 2}$$

$$x=\dfrac{6\pm\sqrt{4 }}{ 2}$$

$$x = \dfrac{ 6 \pm 2}{2}$$

$$x = \dfrac{ 6 + 2 } { 2 }$$   OR   $$x = \dfrac{ 6 – 2}{2}$$

$$x = \dfrac{8} { 2}$$   OR   $$x = \dfrac{4} {2}$$

$$x = 4$$   OR   $$x = 2$$

## EXAMPLE#2:

Solve the quadratic equation $$4x^2 + 12x = 7$$ by the Quadratic formula.

Solution:

First, we equation as: $$4x^2 + 12 x – 7 = 0$$

Now solve the equation here $$a = 4, b = 12, \space \& \space c = – 7$$

$$x = \dfrac{- b \pm \sqrt{b^2 – 4ac}}{ 2a}$$

$$x = \dfrac{ – 12 \pm \sqrt{(12)^2 – 4 (4) (-7) }}{ 2 (4)}$$

# Users Of The Geometric Method To Solve The Quadratic Equations:

Besides Babylonian mathematicians, the mathematicians of China, Egypt, India and Greece also used the geometric method (not in the form of general formula) to solve the quadratic equations only for the positive roots.

# Contribution Of Indian Mathematician:

In 628 AD an Indian mathematician named Brahmagupta explained in wording the following explicit form of formula to solve the quadratic equations as:

$$x = \dfrac{\sqrt{4ac + b^2 } \space – b }{ 2a}$$

# Contributions Of Al Khawarizimi And Other Muslim Mathematicians:

In 9th century an Islamic mathematician Muhammad ibn Musa al-Khwarizmi inspired by the concept of Brahmagupta and developed a set of formula to solve the positive roots of the quadratic equations. He also developed the completing square method and identified that the discriminant should be positive in solving quadratic equations. Another Muslim mathematician Hamid ibn Turk proved the working of the Khwarizmi that the discriminant should not be negative to solve the quadratic equations.

The common thing in the above discussion is that all the mathematicians solved the quadratic equations for positive roots but later on Muslim mathematicians proved that the quadratic equations could be solved for negative roots as well as for irrational numbers.

# Contributions Of Simon Stevin And Rene Descartes:

Finally in 1594 Simon Stevin by his work obtained the formula for Quadratic equation and in 1637 the formula for Quadratic equation which we use today as $$x = \dfrac{ – b \pm \sqrt{b^2 – 4ac }}{ 2a }$$ was published by Rene Descartes in La Geometrie.

So, the above complete history of the quadratic formula is telling us how the quadratic formula derived. The quadratic formula is the best Quadratic equation solver. There are other methods to solve the quadratic equations like completing the square method, factoring method and graphing method etc. but the quadratic formula is widely used by the people of different fields to solve the quadratic equations problems. Now we understand the Quadratic formula’s calculation by the following examples:

## EXAMPLE #1:

$$x = \dfrac{ – 12 \pm \sqrt{144 + 112 }}{ 8}$$

$$x = \dfrac{ – 12 \pm \sqrt{256 }}{ 8}$$

$$x = \dfrac{ – 12 \pm 16 }{ 8}$$

$$x = \dfrac{ – 12 + 16 }{ 8}$$   OR   $$x = \dfrac{ – 12 = 16 }{ 8}$$

$$x = \dfrac{4} {8}$$   OR   $$x = \dfrac{ – 28 } {8}$$

$$x = \dfrac{1 }{ 2}$$   OR   $$x = \dfrac{ – 7 } {2}$$

Solution sets {0.5, – 3.5} Answer.

From the above discussion and Quadratic formula calculation examples it will be clear to you how the Quadratic formula derived? And how you can solve the quadratic equation by Quadratic formula.

If you are a student and in your daily hectic schedule have no time to complete your assignment and you are trying to find the quadratic function calculator to solve your Quadratic equations problems.

So, solve your problems in less time by using our Quadratic function calculator or Quadratic formula calculator and be the best Quadratic equation solver in front of your class.

Besides the quadratic formula calculator, you may also use our factoring quadratics calculator to solve your Quadratic equations problems by factoring method. We not only guide you in solving quadratic equations by different methods but also explain to you the theory of all solving methods of quadratic equations so, understand with us and be a champion of the quadratic equation solver.

# Graph Presentation “Parabola” Of A Quadratic Equation

### Introduction Of The Quadratic Equation:

The second-degree polynomial equation which has only one unknown variable is known as Quadratic equation. The standard form of Quadratic equation is as $$ax^2+ bx+c=0$$ and the formula for Quadratic equation is $$x = \dfrac{ -b \pm \sqrt{b^2 – 4ac}}{2a}$$. New inventions have made our work much easier just like Quadratic formula calculator which is the great Quadratic equation solver just put the values of a, b, & c and get your answer instantly.

### Methods Use To Solve The Quadratic Equation:

There are four methods to solve quadratic equations: Quadratic formula, completing the square method, factoring method, & the graphing. Amazing thing is that there are different Quadratic function calculators available online to solve quadratic equations in less time just like Quadratic formula calculator, & factoring quadratics calculator etc.

If you are not good at factoring and find the quadratic function calculator or find the best Quadratic equation solver then you are at the right place because our factoring quadratics calculator will solve your equations in minutes.

### Uses Of The Quadratic Equation:

The use of Quadratic equation is not only confined with mathematics, algebra, & geometry but in any field like chemistry, physics, daily science, and in accounting if we find any equation which has at least one square value or in Quadratic standard form then we may solve that equation by any of the above-mentioned methods. So, if you belong to any field and want to solve quadratic equations then you may also use Quadratic formula calculator or Quadratic function calculator without searching the formula for Quadratic equation.

### Graphing Or Sketch Parabola Of The Quadratic Equation:

If we solve quadratic equations graphically then it shows u shaped upwards and downwards and this u shaped is known as parabola. In Quadratic equation if the value of a is equal to 1 or greater than 1 then parabola opens upwards and if the value of a is less than 1 then parabola opens downwards as shown in the following: Let’s understand the graphing of the quadratic equation by an example:

### Example To Solve Quadratic Equation By Graph (Parabola):

Solve the quadratic equation by graph

$$y = x^2 – 3x – 1$$

Now we put the values of x to find the values of y:

If $$x = -2, y = (-2)^2 – 3 (-2) – 1 = 9$$

If $$x = -1, y = (-1)^2 – 3 (-1) – 1 = 3$$

If $$x = 0, y = (0)^2 – 3 (0) – 1 = – 1$$

If $$x = 1, y = (1)^2 – 3 (1) – 1 = – 3$$

If $$x = 2, y = (2)^2 – 3 (2) – 1 = – 3$$

If $$x = 3, y = (3)^2 – 3 (3) – 1 = – 1$$

If $$x = 4, y = (4)^2 – 3 (4) – 1 = 3$$

If $$x = 5, y = (5)^2 – 3 (5) – 1 = 9$$

$$X: -2, -1, 0, 1, 2, 3, 4, 5$$

$$Y: 9, 3, -1, -3, -3, -1, 3, 9$$ In the above quadratic equation, we computed the coordinates for eight points. We noted that the plotting of any number of points for which x is less than 3 is not giving a shape of the curve and when the values of x are greater than 3 then it is giving a curve shape. Thus, it is important to choose the values of x for plotting the graph and generally we should determine at least five well-chosen values of x for getting an accurate sketch of a quadratic equation. We can see in the above graph that if the value of a is equal to 1 or a is positive then the parabola is opening upwards.

### Concepts That Are Related With The Graph Of Quadratic Equation:

Now we discuss the following concepts which are associated with the graph of Quadratic equation:

1. Roots.
2. Discriminant.
3. Vertex.
4. X-intercept.
5. Y-intercept.

#### Roots:

The Quadratic formula is also known as the roots of the quadratic equation and the formula for Quadratic equation is as:

$$x = \dfrac{ -b \pm \sqrt{b^2 – 4ac}}{2a}$$

We can write the roots of the quadratic equation as:

Root for $$x = \dfrac{ -b + \sqrt{b^2 – 4ac}}{2a}$$

Root for y $$x = \dfrac{ -b – \sqrt{b^2 – 4ac}}{2a}$$

#### Discriminant:

In Quadratic formula the part $$b^2 – 4ac$$ is known as Discriminant and it is represented by D and the nature of the roots of the quadratic equation depends on the value of the D.

If The Value Of The D = 0:

The roots of the quadratic equation will be equal.

If The Value Of The D Is Greater Than 0:

The roots of the quadratic equation will be real.

If The Value Of The D Is Less Than 0:

The roots of the quadratic equation will be imaginary.

If The Value Of The D Is Not A Perfect Square And Greater Than 0:

The roots of the quadratic equation will be irrational.

If The Value Of D Is A Perfect Square And Greater Than 0:

The roots of the quadratic equation will be rational.

If The Values Of A, B, & C Are Integers And The Value Of D Is Greater Than 0 And Also A Perfect Square:

The roots of the quadratic equation will be integers.

### Vertex:

In a Quadratic equation graph parabola, the maximum and the minimum point is known as vertex. We find the vertex of the quadratic equation by the following formulas:

Vertex for $$x = \dfrac{-b}{2a}$$

.

Vertex for $$y = \dfrac{-b^2 – 4ac}{4a}$$

.

We can also write the above vertex formulas as:

Vertex for $$x = h = \dfrac{-b}{2a}$$

.

Vertex for $$y = k = \dfrac{-D}{4a}$$

.

### X-Intercept:

In the quadratic equation if we want to find the x-intercept then we will put the value of y as 0 and will solve the equation simply by adding and subtracting.

### Y-Intercept:

Similarly, if we want to find out the y-intercept then we will put the value of the x as 0 and will solve the equation in a simple way.

So, if you are new to solving the quadratic equation graphically then from the above discussion you will be able to sketch the parabola of the Quadratic equation. In solving the quadratic equation if you are facing any problem and you are finding the quadratic equation solver and finding the formula for Quadratic equation or trying to solve quadratic equation by Quadratic function calculator then try our Quadratic formula calculator and get flying colors in your assignment.

# Completing The Square Method To Solve The Quadratic Equation

As we know, solving the quadratic equation by using the Quadratic formula is quite easy. But when it is said to solve the Quadratic equation by completing the square method then students get a little bit confused. If you are included in those who are finding Quadratic equation solvers or to find the quadratic function calculator to solve quadratic equations by completing square method, don’t worry we will completely guide you how you will solve these quadratic equations.

### Completing The Square Method To Solve The Quadratic Equation

In completing the square method, we follow the following steps:

#### Step# 1 Transposition The Equation:

First transpose the equation if needed.
OR write the quadratic equation in standard form.

#### Step # 2 Make The Coefficient Of X Is Equal To 1:

Divide the exponent of the x by the whole equation to make the coefficient of x is equal to 1.

#### Step # 3 Make A Perfect Square:

Add the value $$(½ × coefficient of x)^2$$ to each side of the equation to make it a perfect square.

#### Step # 4: Simplify The Equation:

Convert the left side of the equation into square form and simplify the values of the right side of the equation.

#### Step # 5 Take The Square Root:

Take the square root of the both sides of the equation and solve the equation with basic rules like add and subtract then finally after solving the equation we will get the value of x or required solution sets.

Let’s understand the completing the square method to solve the quadratic equations by the following examples:

#### Example# 1:

Solve the equation by completing the square method.

$$X^2 – 2x = 2$$

Solution:

Here we skip the first and second step because the equation is already in standard form and there is no need to make the exponents of x equal to 1. Now let’s start other steps:

Add $$(\frac{1}{2} × b)^2 = (\frac{1}{2} × -2)^2 = (-1)^2 = 1$$ on the both sides of the equation,

$$X^2 – 2x + 1 = 2 + 1$$

$$(x-1)^2 = 3$$

Now taking square root on both sides of the equation,

$$\sqrt{(x-1)^2} = \sqrt{3}$$

$$x-1 = \pm \sqrt{3}$$

$$x = 1 \pm \sqrt{3}$$

Solution sets {$$1 – \sqrt{3}$$, $$1 + \sqrt{3}$$} Answer.

#### Example# 2:

Solve the equation by completing the square method:

$$12 = 4x + 5x^2$$

#### Solution:

By transposition the equation we have,

$$5x^2 + 4x = 12$$

Now divide the exponent of x by the whole equation to make it equal to 1,

$$\dfrac{5x^2}{5} + \dfrac{4x}{5} = \dfrac{12}{5}$$

$$x^2 + \dfrac{4x}{5} = \dfrac{12}{5}$$

Add $$(\dfrac{1}{2} × coefficient of x)^2$$

We have $$(\dfrac{1}{2} × \dfrac{4}{5})^2 = (\dfrac{4}{10})^2 = 16/100$$

as $$\dfrac{16}{100}$$ cuts by the table of $$4 = \dfrac{4}{25}$$

Add $$\dfrac{4}{25}$$ on both sides of the equation,

$$x^2 + \dfrac{4}{25}x + \dfrac{4}{25} = \dfrac{12}{5} + \dfrac{4}{25}$$

Note:

At the left side of the equation the square term is $$x^2 + \dfrac{4}{25}$$ we can write it as

$$(x + \dfrac{2}{5})^2$$ Now it is as $$(a + b)^2$$ so, we apply the formula $$a^2 + 2ab + b^2$$ and here $$a = x$$ & $$b = \dfrac{2}{5}$$.

$$x^2 + 2(x)(\dfrac{2}{5}) + (\dfrac{2}{5})^2 = \dfrac{12}{5} + \dfrac{4}{25}$$

$$(x + \dfrac{2}{5})^2 = \dfrac{12}{5} + \dfrac{4}{25}$$

$$(x + \dfrac{2}{5})^2 = \dfrac{60 + 4}{25}$$

$$(x + \dfrac{2}{5})^2 = \dfrac{64}{25}$$

Now taking square root on the both sides of the equation,

$$\sqrt{(x + \dfrac{2}{5})^2} = \sqrt{\dfrac{64}{25}}$$

$$x + \dfrac{2}{5} = \pm \dfrac{8}{5}$$

$$x + \dfrac{2}{5} = \dfrac{8}{5}$$ OR $$x + \dfrac{2}{5} = -\dfrac{8}{5}$$

$$x = – \dfrac{2}{5} + \dfrac{8}{5}$$ OR $$x = – \dfrac{2}{5} -\dfrac{8}{5}$$

$$x = \dfrac{-2 + 8}{5}$$ OR $$x = \dfrac{-2 – 8}{5}$$

$$x = \dfrac{6}{5}$$ OR $$x = \dfrac{-10}{5}$$

$$x = 1.2$$ OR $$x = -2$$

So, the solution sets {1.2, -2} Answer.

The use of Quadratic equations is not only confined with mathematics, algebra, and geometry. It is widely used in physics, chemistry, business, sports, law & management etc. The graph of Quadratic equation is known as a parabola and the amazing thing is that anything which we see in our daily life looks like a parabola then we can apply the quadratic equation.

For instance, we can find the height of a ball if a batsman hits it from ground and can find the speed of a car etc. The Quadratic equation is solved by quadratic formula, completing the square method, graph, and factoring method. We can use any one of the methods. If you are the chemistry or engineering student and finding the quadratic function calculator or finding factoring quadratics calculator then you are at the right place because our factoring quadratics calculator or our Quadratic function calculator will solve all your Quadratic related problems.

In earlier days we used calculators for adding, subtracting, multiplying, and division only but in this scientific era all other calculations we can perform in seconds through special tasks calculators just like quadratic formula calculator.

If you are in the class and in a hurry to solve quadratic equations and trying to be a quadratic equation solver, don’t waste your time just put your values in our Quadratic formula calculator and get your answer quickly. In the Quadratic function calculator, we don’t need to put the formula of the Quadratic equation.

If you are looking for a formula for Quadratic equation to solve your equation then don’t need to search the formula for Quadratic equation because in our Quadratic function calculator just put the values of a, b, & c and get the instant accurate result.

When we get introduced by the basics of algebra then we learn about base, exponent, term or order etc. After the clarity of basics, we learn about the degrees of polynomials. As we know that polynomials are based on variables, constants, and exponents. The exponents of the variables of the polynomials are based on positive integers. The polynomials are separated by plus or minus sign and each separated polynomial is known as a term. If the polynomial is based on one term is known as monomial, if based on two terms is known as binomial, and if it is based on three terms is known as trinomial. Now we know how to determine the degree of a polynomial; it is very simple. First of all, we separately add up the exponents of each term and then will select the highest one which will be called the degree of polynomial. After a little introduction about the polynomial, we come to our targeted topic, the quadratic equation. The word Quadratic is derived from a Greek word Quadratum which means double. The quadratic equation is based on the second degree of the polynomial because at least one variable in Quadratic equation is squared. The standard form of the Quadratic equation is $$ax^2 + bx + c = 0$$, where x is an unknown variable, a, b, & c are any numbers or constants and a is not equal to zero.

The formula which we use to calculate the quadratic equation is as:

$$x = -b \pm \dfrac{\sqrt{b^2 – 4ac}}{2a}$$

Example:

$$2x^2 – 8x – 24 = 0$$

In this Quadratic equation a = 2, b = – 8, & c = -24. Now apply the formula:

$$x = -(-8) \pm \dfrac{\sqrt{(-8)^2 – 4(2)(-24)}}{2(2)}$$

$$x = 8 \pm \dfrac{\sqrt{64 + 192}}{4}$$

$$x = 8 \pm \dfrac{\sqrt{256}}{4}$$

$$x = 8 \pm \dfrac{16}{4}$$

$$x = 8 + \dfrac{16}{4}$$,$$x = 8 – \dfrac{16}{4}$$

$$x = \dfrac{24}{4}$$,$$x = \dfrac{-8}{4}$$

$$x = 6$$,$$x = -2$$

Thus, solution sets [ 6, – 2] Answer.

If you are included in those persons or students who are finding the formula for Quadratic equation then our Quadratic formula calculator is here to solve your problem.

This era is of modern technology and everything is going to be digital and is easier to use. Our Quadratic formula calculator or Quadratic equation solver calculator is also very simple and easy to use so save your time and increase your marks just put the values and get the accurate result.

### Methods To Solve Quadratic Equation

There are four methods to solve the quadratic equation:

• Completing the square.

• Graphing

• Factoring

We have discussed the quadratic formula above then now we will just discuss here the most important factoring method to solve the quadratic equation. In the factoring method the quadratic equation is solved in this way that first we standardize the equation then factorizes the non-zero side and finally set each factor to zero. After this process we get the equation then we solve it easily and find the solution sets for the variable X.

In the factoring method it is essential to understand the factorization first but if you have a problem in understanding factorization and you are finding a factoring quadratics calculator for help, our factoring quadratics calculator is waiting to solve your Quadratic equation by factoring method.

### Are You Finding The Quadratic Function Calculator?

Our aim is to assist you in understanding the quadratic function and to solve quadratic equations easily. If you try to find the quadratic function calculator to get accuracy in your results then try our Quadratic function calculator because your problem will be solved here in less time with accuracy.

# Airplanes & Mobile Phones – Real World Examples Of Quadratic Equation

From mobile phones to airplanes, engineers and mathematicians are still using the quadratic equation solver calculator to solve complex problems. Amazing fact is that it is not necessary that you learn the Quadratic equation formula or to find out the formula for Quadratic equation the quadratic formula calculator automatically solves the problem by entering the values.

To solve quadratic equations other methods are also used like graph method, factor method etc. The factor method is a famous one and there is a need to understand the factorization principles. But the factoring quadratics calculator has made this task simple and easy. By the factoring quadratics calculator, we can solve the quadratic equations in minutes.

The story of the quadratic equation started a long time ago when the Babylonians introduced the tax system to the farmer. Understandably, the farmers were not happy about the tax so they had to calculate the land needed to be cultivated to make up for the loss in taxes. They came up with the quadratic equation to make up for their loss and estimated the land on which they had to grow crops in order to make the ends meet. As ‘necessity is the mother of invention’, the solution to a simple farming problem is still being used to solve greater problems.

There are hundreds of uses of quadratic equations which we observe in daily life and anyone may find the quadratic function calculator to resolve the problems. What we lack is the understanding that this equation can help us estimate the results or simply understand the phenomena. Today, I have compiled a list of practical uses of quadratic equation so the next time you are solving it – you know where it can be used.

### Quadratic Chaos – When Math Meets Biology

Don’t be surprised if any medical student finds the quadratic function calculator because the quadratic formula calculator also helps in solving biological problems. The concept of quadratic chaos is defined by the quadratic equation with the focus on biology. The population of a specific species can be estimated accurately with the help of quadratic equations. There is no pattern to the growth of species hence it is considered as chaos. In this case x is the population while ‘n’ is the years so xn+1 is considered to be the sub function of xn. If axn was bred successfully while bxn2 dies due to overcrowding – the equation will be,

xn+1=rxn(1–xn)

The focus of the equation is not just estimating the growth – it is more about understanding and evaluating if the prediction is accurate or hopeless.

### Mobile Phone and Complex Quadratic Equation

The imaginary numbers were first discovered trying to find an answer to the cubic equation actually. When the imaginary numbers were used to solve the cubic equation, mathematicians were actually able to find the actual results. It was important to justify this mathematical fix otherwise we would have kept on introducing new letters ultimately running out of letters. Thus, complex numbers were introduced which are the combination of numbers and alphabets. The incorporation of cubic equations with complex numbers ultimately led to solving quadratic equations that helped in predicting electron motion. It also assisted in designing the integrated circuit that can perform complex tasks. These circuits are still being used in almost every technological invention including DVDs, computer, cars, and mobile phones. Phone signals convert the speech into high frequency radio waves and this behaviour was understood by the quadratic equation. You can say that without the simple quadratic equation ‘x2 = -1’, we would not have invented mobile phones!

Whether we think about simple machines or the more advanced one – one thing is very clear; it wouldn’t have been possible without calculation. We owe almost all of our invention to mathematics. From growing food as per the need of the population to trying to predict natural phenomena from observing the movement of galaxies to stepping foot on moon, from calling our loved ones thousand mile away to predict the spread of infection – everything became possible for us when we dived into the realm of mathematics and its branches. One thing is very clear here – if we want to progress further, we need to reintroduce this subject to our students. Making the student understand the concept and logic behind the mathematical equation is the only way forward and we need to focus on it immediately. Maybe we will get lucky and find the next Einstein or Newton!

# Mathematical Equations That Remarkably Impacted The World

Calculation, equation, and, math is continuously revolutionizing our world. From the time mankind wanted to calculate the field area for growing crops – there was a thirst to know and understanding the secrets of the world. Why apple always fall down rather than flying, is there a pattern to the movement of star, what can assist in navigation, and why birds fly while we cannot – these questions of the curious minds lead to the thirst of known and the answer provided the mean to modernize the world one invention at a time!

1- Calculus

Due to the applicability of calculus, it is not only used in mathematics but in engineering biology, physics, chemistry, and many more branches of science. Calculus can help you in the determination of weather pattern movement of sound, movement of light, and motion of astronomical objects.
Euler’s Polyhedra Formula
Fourier Transform

2- Law of Gravity

Gravity is an undeniable force responsible for the existent of our planet. The law of gravity helps in the evaluation of weight and speed leading to significant modernizations including race car amd airplanes.

3- Logarithms

There are many example of the use of logarithms in the real world starting from interest rate to Google page rankings. Logarithms are also used to detect changes in multiplication and help count them.

4- Maxwell’s Equations

It is the set of 4 differential equations that describe the relation between electricity and magnetism. These equations are the basis for understanding the behavior of electromagnetism. From MRI scanners in the hospital to computer – the credit goes to the basic understanding of Maxwell’s equations.

5- Navier-Stokes Equations

These differential equation helped us understand the behaviors of flowing liquids such as smoke rising from cigarette, water moving through pipes, and air flow over plane wings. Navier-Stokes Equations are also used to model the weather and observe ocean currents.

6- Normal Distribution

Normal probability also known as normal distribution forms a bell curve and it is signifint in statistics. It is used in social sciences, physics, and biology to define the behavior of large groups of independent processes. Normal distribution is followed in the measurement of errors, heinght, IQ score, and blood pressure.

There are various functions that are modeled by quadratic equation including shooting a cannon, hitting golf ball, and diving. You can calculate the expected profit you are going to get if you are using the quadratic equation. It can prevent unwanted surprises and provide you with the accurate numbers and what to expect in the future. Even in the business where you are simply selling bottled water, it can help you estimate how many bottles you have to sell to generate the profit you want.

8- Relativity

Relativity opened the door to understanding – be it our understanding of the outer space or the speed of light. It provided us with the idea that light speed is universal but the time factor is different for the speed of people or objects. Relativity helped us understand the fate, structure, and the origin of the universe.

9- Schrodinger’s Equation

The behavior of atomic and subatomic particle is defined by Schrodinger’s Equation. It enhance the understanding of quantum physics hence played a huge role in the development of computing devices. Computational chemistry is the direct application of Schrödinger equation and it is currently being used in medication and engineered food.

10- Second Law of Thermodynamics

According to the second law of thermodynamics, heat flow from hot to cold environment due to the change in temperature. This is the concept used in the working of internal combustion engines used in airplane, ship, car, and motorcycles. The law is applicable to all engine cycles and led to the progress of modern vehicles.

11- The Pythagorean Theorem

Whenever you need to find out if a triangle in acute, right-angled, or obtuse – you can use Pythagoras theorem for that. It made the life of mathematicians easier as it help them to find the missing length of any side of a triangle.

12- The square root of -1

The square root of -1 = I, this process gave rise to complex numbers that are supremely elegant. In case an equation have complex number solution, it will represented by ‘I’. With the help of this equation, mathematicians were able to find symmetries and the properties of the number which are implemented in signal processing and electronics.

13- Wave Equation

Wave equation as the name indicates describe the behavior of waves along with ripples, guitar strings, and incandescent bulb light. It is one of the first differential equations that helped us understand other differential equations as well.

The world of mathematics is abundant with equations that helped us revolutionize the world as we know it today. We were not only able to understand the concept behind natural phenomenon ut also manipulate them for the modern advancement. These were just the few examples, stay tuned to know more!

# Mathematicians Who Changed The World

### 9 Mathematicians Who Changed The World

Mathematicians are the true hero but you don’t really know about them. I can recall two or three mathematician and that’s too because they were famous physicist. Yes, I am talking about Newton and Einstein! If you ask me to list 5 mathematician, I will fail and not the normal fail where you get one name wrong – I will get all the names wrong apart from these famous too. If mathematicians are heroes then why don’t we hear about them? Well the answer to this question is pretty simple – because most of us hate math or just not bother with it! It is a sad reality especially when you look around yourself and get to know that almost everything in your room include some kind of math or any calculation. You are bored and browsing on the internet – math. You want to get the accurate dimension of the room for a new carpet – math. The circuit, the bulb, the plug, and even the spider web on you wall have some kind of mathematical pattern or logic behind it.

We appreciate the glam of movie star but forgets to appreciate the inventors of camera. We love animated movies but forget to acknowledge the efforts of those who made is possible. This articles is an effort to appreciate our mathematician so if someone ask you to name a few with their invention – you know the answer!

1- Albert Einstein

He doesn’t need any introduction – he is probably the most known mathematician. Even though his contributions in the field of physics are notable and famous, his mathematical achievement are often not discussed. He contributed multiple equations to geometry and calculus, 10 of which are known as the ‘Einstein Field Equations’ published in 1915. He provide that how stress-energy can impact the curvature of space-time.

2- Carl Gauss

Carl Gauss was born in 1777 in Germany. Even though he was born in a poor family, he soon started to show the signs of a brilliant mind. He started elementary school at the age of 7 and his teacher was amazed when he summed the integers from 1 to 100 just by spotting that there were 50 pairs each summing 101. He was the discovered of Bode’s law, arithmetic – geometric mean, prime number theorem, binomial theorem, and law of quadratic reciprocity – all that while he was in college! He published his work title as ‘Disquisitiones Arithmeticae’ which provided the construction of a regular 17-gon nu ruler and compasses.

His second book was ‘Theoria motus corporum coelestium in sectionibus conicis Solem ambientium’ which was published in 1809. It provided the accurate description of the motion and movement of the celestial bodies. He also discussed conic sections, differential equations, and elliptic orbitals in the first part of his book. The second part was focused on the refinement of estimating planet’s orbitals.

3- John von Neumann

John von Neumann was born in Budapest in early years of 20th century and you can call it our luck as her designed the architecture served as the base for the development of computer. His work was so accurate that even today the same architecture has been used with minor alterations or improvements. Even the most advanced computers of this days are cycling through a series of basic steps over and over again. This is the processing behind playing video or doing literally anything on the computer and it was first proposed by John von Neumann.

He hold two Ph.D. one in mathematics and second in chemical engineering just because his father wanted it. He worked with Albert Einstein at Princeton University and made discoveries in quantum mechanics, computer science, set theory, game theory, and geometry.

4- Alan Turing

Alan Turing is known as the father of computer science and he was a British mathematician. He is well known for breaking Nazi crypto code during World War II. The code was protected by Enigma machine and breaking it gave the Allies advantage over Nazis. Some even say that it is one the main reason the Allies were able to defeat the Nazis.

Besides doing this huge favor to the world, Turing set the basis for the development of modern computer. Till this day computer operate similarly to the ‘Turing machine’ designed by Alan Turing. Turing test is used to evaluate how well an Artificial Intelligence (AI) programs works. Any program that can pass the Turing test can have a conversation (chat) with a human fooling him to think that he is chatting with a human.

His name is still a prestigious one in the field of computer science as Tuning Award is named after him – which is equivalent to Nobel Prize in Chemistry.

5- Benoit Mandelbrot

Benoit Mandelbrot discovered fractal geometry based on the complex shape built on simple formula. They are the basis of computer animation and graphics so you can think why Benoit’s name is included in this list. Without fractal much wouldn’t have been possible as they are also used to design computer chips and cellphone antenna.

These are just the few name who made it big in the field of mathematics. There are hundreds of other mathematicians who are neglected and their work is still being used in different sector. It is important that we know our ancestors and the effort they put into making this word an advanced place!

# From Father of Math to Father of Algebra

Mathematics is a bit dry for me. I start to solve an equation and suddenly the lizard on the wall looks interesting as compared to the equation I am supposed to solve. My lack of focus is the only thing that made me research the importance of the subject more. The more I got bored, the more I tried to understand the reason behind it. The question I asked myself was why I am solving this equation followed by who came up with this subject, and are there any mathematicians who got famous doing math? Well, as you have guessed – one question lead to another and soon I was searching for famous mathematicians. Turned out there is not one but hundreds of famous people who are mathematician. This article is a tribute to those who (unlike me) were brilliant at math. They not found the value of their ‘x’ but help millions to find it too. So let’s dive into the realms of mathematics and recognize the efforts of notable mathematicians who shaped the world as we know it today!

1- The Pythagoreans

Perhaps the first known mathematician and his students known as the Pythagoreans and let me make this clear – they were obsessed with number. Pythagoras presented a theorem about right angled triangles ad still considered to be the Father of Geometry for this feat. Number was not only a subject for the Pythagoreans – it was the spiritual qualities of numbers that made it the focus of effort.

2- Euclid

Euclid was a Greek mathematician and wrote a book titled ‘Elements’. It was the primary textbook till 19th century. He was the brain behind Euclidean Geometry which provide the basis for modern geometry. He was also the first person to formalize mathematical proofs as the method of exposition.

3- Archimedes

Even though the achievements of Archimedes are not limited to the field of mathematics he thought that it is the only field worth pursuing. He is known as one of the greatest mathematicians as he perfected the integration methods helping him find the surface area, volume, and center of mass of the geometrical figures. He also gave us the accurate approximation to pi and help us find the mass of object immersed in liquid.

He is often known as the grad father of computer as he proposed the concept of algorithm which was later used in the development of first computer. We owe our understanding of algebra to Musa al- Khwarizmi. He also worked on solving quadratic equations, surveying plot of land, and dividing up inheritance.

5- John Napier

John Napier is known for the invention of logarithms which are still being used in different mathematical calculations. He was the first mathematician who worked towards bringing the decimal point into common usage. His logarithms helped with trigonometric calculations in astronomy and navigation.

6- Johannes Kepler

With German roots, Johannes Kepler was the first mathematician and astronomer who provided the idea that earth and other planets revolve around the sun in elliptical orbitals. He was also the first mathematician to provide the mathematical treatment of the close packing of equal spheres that explained the shape of honeycomb cells. He also provided a method to find the volume of solids and contributed to calculus.

7- Rene Descartes

The credit of using alphabets in math goes to Rene Descartes. He is known as the ‘Father of Analytical Geometry’ because of the discovery of plotting two- dimensional points on a mathematical plane. Descartes also developed four rules for deductive reasoning which are not only applied to math but to other fields of study as well.

8- Blaise Pascal

He is famous for Pascal’s triangle and the number used in this triangle are binomial, each number is the sum of number above. After coming out with Pascal’s triangle, he presented the probability theory. Pascal was also the first mathematical to experiment with the atmospheric pressure and came up with the statement that vacuums exit in the real world too.

9- Isaac Newton

Apart from his outstanding research in the field of physics, he is known as the ‘Founder of Calculus’. Newton not only gave us three laws of motions but also discovered colored spectrum along with the discovery and calculation of gravity ultimately helping us to calculate the mass of an object.

10- Gottfried Wilhelm Leibniz

The credit of discovering binary numeral system or the base-2 system goes to Gottfried Wilhelm Leibniz. The binary system is still used in computers and related devices ascertaining that Gottfried’s work was value in the development of compute – the most widely used device nowadays. Gottfried Leibniz discovered infinitesimal calculus which is focused on differentiation, integration, and limits of function.

11- Thomas Bayes

He was the first mathematician to use probability inductively along with the establishment of mathematical basis for probability inference (calculating the frequency considering the past trials). Bayes’s theorem is also a significant achievement of Bayes as it can find probability of an event in past or future based on the pattern of available data.

12- Leonhard Euler

Leonhard Euler worked in different fields including mathematics, physics, astronomy trigonometry, geometry, algebra, and calculus. He introduced the notation for a function f(x). He also popularized the use of the Greek letter π to denote the ratio of a circle’s circumference to its diameter.

13- William Playfair

Playfair was one of the most important developer of statistical graphics focusing on the idea that chart and graphs can represent data more effectively. He devised and published pie charts, statistical line graph, and the bar chart. He provided a common language to data representation.

These were some of the notable mathematicians who changed the society as we know it. Do you know any other – tell us in the comment section so we can learn about them too!

# Asians Are Good At Math – Reality, Myth, Or Plain Racism?

From mathematic to biology, from chemistry to physics, from theoretical to practical interventions – Asians are making their mark and it’s high time you start thinking how? Ever noticed a pattern in the results of any mathematic contest? Ever wondered why there are so many research papers being submitted by Asian countries? No, well you are not the first one to ignore this. We have been so used to celebrate the inventions of western scientist that we got used to the idea that all inventions are western.

Most of the time we ignore the fact that we are not making as much progress in the scientific world as our Asian counterparts and this is base of our problem. What could have been a healthy competition has turned into blatant ignorance. In case the achievement of the Asian professionals are highlight – they are mostly tainted with racism. Yes – I final said the word. With the recent stark evidences, we cannot ignore the fact that we still don’t have a cure for racism in our society.

Let me give you a recent example. The cases of Covid – 19 were first reported in Wuhan, China. It shake the basis of world economy. China was the first country to report the virus and the first to control it. Now as we are dealing with the consequences associated with the spread of Covid – 19 such as loosing job, poverty, deaths and much more – how many of us are thinking about how china was able to control it. Let’s just think about it, with the hub of virus, the largest population of the world was able to defeat a virus they don’t know much about. The answer to this question in strict control as well as intensive research. With the threat came the approach to solve it, from total shut down to multiple therapeutical approaches – China defeated the virus while we are still living under the protection of masks and calling Covid – 19 the Chinese virus.

When we go through the scores of different countries in math contests – Asians are performing better. They are academically better and have a stronger grasp of the concept. The reason of this outstanding performance if not limited to a single factor rather it is a combination of various factors that work together to make Asians better. The first factor is old and simple hard work – Asian are known to be hard worker. With centuries of oppression and being undervalued, they had to work extra hard to make their make and thus they embrace the quote that ‘hard work pays off’. In most of the international tests, a student have to fill a detailed questionnaire and by detailed I mean around 100 questions, with persistence and hard work Asians attempt those questions while western student gets bored and leave the questions unanswered. This is a clear example of the importance of persistence, concentration, and hard work in action.

The second factor is the parental involvement in the choice of academic career. Even though there is an ongoing debate about different parenting styles, it seems that Asians are doing something right! They have a significant role in the academic choices of their children. With the efforts of first few years when parent are teaching their children the basis of academics like alphabet and counting, they shape the mind of the child to put extra effort in the academics.

Another reason of Asians better in math is based on simple logic – they don’t get as much vacations as we do. A Japanese school year us 243 days long, the South Korean is 220, while the average American school years is just 180 days long. With longer school years, there is increased chances of learning something new. Even if the children are not learning anything new, they are revising what they have learned previously and hence they are getting better and better every day & there are more than 50 days of getting better each year!

Another reason for Asians being good at math is their numbers are easier. Take the example of this sequence, 4, 8, 5, 3, 9, 7, 6. Human have a 2 second span of memorizing the digits. If you speak English you are going to recall this number sequence right 50% of the time – if you are Chinese, you are going to get it right every time.
yī = one
èr. = two
sān. = three
sì = four
wǔ = five
liù = six
qī = seven
bā = eight

As we most easily memorize whatever we can say or read within that two-second span, you can learn the digits easily. Chinese digits are brief so they fit into 2 second memory loop and help you get them right every time! These are the facts that make Asian better in math and other subject. So the next time you want hear someone throwing a racist statement away at Asian – tell them what they are doing wrong and how they can get better too. After all tolerance and information is the only way out of this dire times.