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.

QUADRATIC EQUATION

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.

“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 \(a=1,b=-6, \space \& \space c=8\)

Now apply the quadratic formula:

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

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

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

\(x=\dfrac{6\pm\sqrt{4 }}{ 2}\)

\(x = \dfrac{ 6 \pm 2}{2} \)

\(x = \dfrac{ 6 + 2 } { 2 } \)   OR   \( x = \dfrac{ 6 – 2}{2}\)

\(x = \dfrac{8} { 2} \)   OR   \( x = \dfrac{4} {2} \)

\(x = 4 \)   OR   \( x = 2\)

Solution sets {4, 2}. Answer

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 \(a = 4, b = 12, \space \& \space c = – 7 \)

Apply the quadratic formula:

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

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

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.

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.

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.

Uses Of Quadratic Equation

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.

Quadratic Formula Calculator

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.

Formula For Quadratic Equation:

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.

Quadratic Formula Calculator Or Quadratic Equation Solver Calculator

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:

• Quadratic formula

• 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.