Quadratic Equation Solver

In the present day, we are entitled to discuss something out of Mathematics, the Quadratic Equation. Although, this is a famous subject matter amidst different exams and studies, it requires sufficient amount of practice, subsequently. Students often get confused and relate quadratic equations with linear equation. Let me make it clear, my patrons, it is entirely distinct from the Linear Equations.

The main difference among these equations is the latter will definitely comprise of a term ‘x2’. The important fact to keep in mind is, a term with power 2 or a term with a degree 2 in an equation makes it a quadratic equation. This article further discusses the significance of quadratic equations and its value in the colossal of mathematics.

Analysis

A quadratic formula is significant to resolve a quadratic equation, in elementary algebra. Even though, there are various other methods to solve the quadratic equation, for instance graphing, completing the square, or factoring; yet again, the most convenient and easy approach to work out these quadratic equations is the quadratic formula.

The most common form of a quadratic equation is:

\( a x^2 + bx + c = 0 \)

Here a, b and c are constants and x denote to be undetermined with a not equal to 0. A quadratic formula thereby, satisfies a quadratic equation by placing the former into the latter. Considering the above parameters, the quadratic formula may signify as,

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

It is important to learn by heart, that every single solution conferred through the quadratic formula is considered to be a derivation of the quadratic equation. According to the rules of Geometry, this derivation certainly denotes the values for ‘x’ where any parabola clearly denoted as y = ax2 + bx + c intersects the x-axis. This formula engenders the zeros of any parabola and further yields the axis of symmetry of that parabola. Moreover, this formula is essentially used to analyze the number of real zeros present in the quadratic equation.

Let us discuss here the Quadratic Equation Solver:

It is often admitted by the mathematicians though, that the quadratic formula is chaotic and prodigious, at times. Make sure, you are not confused or distracted from the persona of this very important equation going through a horde of numbers, letters and signs.

What Importance Does a Quadratic Equation Hold in Your Real Life?

As a student you might need it in various topics regarding mathematics. Also, you shall make use of this equation in subjects like engineering and physics. There are a few steps, if followed pertinently, the factor of being distracted for the students might lessen to some extent.

Implication of The Quadratic Formula

The variable ‘x’ be compatible with a particular horizontal line on the graph paper. Whereas, the ‘y’ is certainly comprehended to be consistent with a vertical line on the graph, valued ‘0’.

Working Out the Quadratic Formula

Quadratic equations are significantly solved via quadratic formula, and is considered among the top five formulae in the subject of mathematics. It is not important for the students to learn the formulas thoroughly and commit to memory. However, this one is quite resourceful and it is recommended to learn it by heart, not only how to derive it but also how to make use of it.

The Quadratic Formula

The formula may at times look intimidating though, it is useful for you as a student in many areas of your life, nevertheless, you will swiftly be acquainted with it. So, start practicing it as soon as possible.

Solving a Problem Using the Quadratic Formula

The first step to be followed, should be to recognize the values for a, b and c, that are said to be the coefficients. Be confident, and verify if the given equation is in alliance with the basic quadratic formula, i.e., \( a x^2 + bx + c = 0 \):

\( x^2 + 4 x - 21 = 0 \)

  • Since, ‘a’ is said to be the coefficient against x2, for this reason, a = 1;
    Remember that ‘a’ is never equal to zero, owing to the significance of x2 it becomes a quadratic.
  • ‘b’, on the contrary, stands in front of ‘x’ as a coefficient, which makes b = 4.
  • A constant is something that stands individual in the equation and do not hold any ‘x’ alongside. For this reason, here c = - 21.
\( x = \dfrac{ -4 \pm \sqrt{16 - 4 \cdot 1 \cdot (-21)}}{ 2 } \)

solving this looks like:

\( x = \dfrac{ -4 \pm \sqrt{100}}{ 2 } \)

\( x = \dfrac{ -4 \pm 10}{ 2 } \)

\( x = -2 \pm 5 \)

Therefore \( x = 3 \) or \( x = -7 \)

Now, let us integrate the values of ‘a’, ‘b’ and ‘c’ in the formula, derived previously.

Comprehending the Solution:

The problem has two solutions and they both demonstrate the intersecting points of the equation, that is, the x – intercept, the point where the x-axis is crisscrossed by a curve. Whilst, preparing a graph the equation \( x^2 + 3 x - 4 = 0 \), can be viewed as:

Graph of quadratic equation

This graph clearly visualizes the solutions derived from a quadratic formula and the intersecting points are denoted as x = - 4 and x = 1.

Subsequently, it is now easier for students to make use of factoring, graphing or completing the square for solving a quadratic equation. So, possibly the formula is not needed anymore. It is not the case, though, every now and then you may come across a harder quadratic equation which is not easy to understand and solve as shown in the previous example.

Problem Solving – An Alternate Approach.

It is not possible to factorize every equation, each time. For this instance, let us unveil an alternate approach for such equations:

\( 3^2 + 6 x = - 10 \)

Before we shall proceed further, let us make this equation compliant with the original formula. As, all the terms in this equation are set on the left:

This formula gives out an elaboration, that:

\( x = \dfrac{ -6 \pm \sqrt{6^2 - 4 \cdot 3 \cdot 10}}{ 2 \cdot 3 } \)

\( x = \dfrac{ -6 \pm \sqrt{36 - 120}}{ 6 } \)

\( x = \dfrac{ -6 \pm \sqrt{84}}{ 6 } \)

It is obvious that, one can not acquire a square root for the numbers with a negative sign, this has to be done by presuming imaginary numbers, indeed. This conclusively, tells us that this equation has no real solution. Moreover, it illustrates that at no point in the graph ‘y = 0’; the function will not intersect x – axis, this outcome can also be viewed as similar, when graphed on a calculator.

We know you can’t take the square root of a negative number without using imaginary numbers, so that tells us there’s no real solutions to this equation. This means that at no point will y = 0, the function won’t intercept the x-axis. We can also see this when graphed on a quadratic equation calculator, as seen in the image below:

Quadratic Equation Graph displayed on Calculator