Quadratic Equations

How to Solve Quadratic Equations?

Quadratic equations are the equations that are often called second degree. It means that it consists at least one term which is squared. Because of this reason, it is called “quad” meaning square.

The general form of quadratic equation is ax2+bc+c=0, where a, b and c are numerical coefficients or constants, and the value of x is unknown. One fundamental rule is that the value of a, the first constant, never can be zero. 

These equations constitute a significant part that is necessary to solve several kinds of complicated mathematical problems. In real life, they are used extensively calculating areas, speed and other dimensions of athletics and sports. 


Basic Concepts

There are several ways in which you can calculate quadratic equation. However, two of them are the most useful and accessible. 


Standard Equation

Using this method, you can solve any quadratic equation. Suppose the equation is ax² + bx + c = 0, hence the value of x will be x = \[\frac{-b \pm \sqrt{b^{2} - 4(a)(c)}}{2a}\]. It is also known as the Sridharacharya formula. 

Employing this technique, you will get two types of value. One of these will be positive, and the other one will be negative. 

Example -

Solve this equation,

 x² – 2x – 6 = 0

The given equation is in standard form. Therefore, you can see that, a = 1, b = -2, c = -6. Now you can put the values accordingly. The solution will be

x = \[\frac{-(-2) \pm \sqrt{(-2)2 - 4(1)(-6)}}{2(1)}\]

x = \[\frac{2 \pm \sqrt{4} + 24}{2}\]

x = \[\frac{2 \pm \sqrt{28}}{2}\]

x = \[\frac{2 \pm 2\sqrt{7}}{2}\]

x = \[1 \pm \sqrt{7}\]

Hence, we will get two values from the equation (\[1 + \sqrt{7}\]) and (\[1 - \sqrt{7}\]).


Factorising Method

This method is in quadratic equation class 10 syllabus and therefore essential for you to learn about this in details. 

In this method, you obtain the solution factoring quadratic equation terms. However, for this, the equation has to be eligible for factoring. This technique is easier than others. 

Consider this example of quadratic equation and find the solution. 

 x² -5x + 6 = 0

The equation is the standard form quadratic equation. To find the solution of it, first you have to consider two terms that are b and c. In this case, b = -5 and c = 6. Now you have to find the product of which two numbers will be 6. Also, the sum of the two numbers has to be -5.

For that, find the factors of 6, which can be 1, 2, 3, and 6. Thereby, you will see that the product of 2 and 3 is 6 whereas the sum of 2 and 3 is 5. Since we have to find -5, we have to take -2 and -3, only then the sum will be -5 and product will be 6.

Therefore, two factors of the equation are (x + 2) and (x + 3).


Quadratic Equation Problems

Following is the quadratic equation with solution

3x2 - x = 10

 3x2 - x - 10 = 0

 3x2 - 6x + 5x - 10 = 0

 3x (x - 2) + 5 (x - 2) =0

 (x - 2)(3x + 5) = 0

Therefore, x - 2 = 0, x = 2

And when 3x + 5 = 0; 3x = -5 or; x = -5/3

 Thus, x= -5/3, 2

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FAQ (Frequently Asked Questions)

1. How to Find Roots Quadratic Equation Using Factoring?

Ans. Using the following steps, you can easily solve quadratic equation by factoring. 

First of all, keep all the terms on the left side of the equal sign, leaving zero on the right side. Then, set each factor= 0. Start solving individual equation. Once you get the answer, put the values in the equation for crosschecking. 

For example, 

x² – 6 x = 16 

It becomes x² – 6 x – 16 = 0

Factors are.

(x – 8)(x + 2) = 0

Setting each factor =zero, 

We get, x - 8 = 0; x + 2 = 0

Therefore, x = 8; x = -2

To check, 

8² - 6(8) = 16 or (-2)² -6(-2) = 16

64 - 48 = 16    4 + 12 = 16

16 = 16    16 = 16

Answer: 8 and –2 both these values are solutions of the original equation.

2. Who Invented Quadratic Equation?

Ans. Often it is claimed that Babylonians first solved quadratic equations about 400 B.C. however, they did not have the idea of an equation, and instead, they used to employ an algorithm to solve mathematical problems. 

Also, it is believed that later in 300 B.C. Euclid and Pythagoras developed an approach to solve these equations. They used to follow this method to find the length of an object, that is now presumed as the root of a quadratic equation. 

Brahmagupta in India also gave a modern method to solve this equation. Therefore, it can be said that the contribution of all these legends gave birth to the modern version of quadratic equation.

3. What is Quadratic Formula and Where it is Used?

Ans. Quadratic equation formula is a method to solve quadratic equations. However, there are other methods as well to solve such kind of equations. One of the significant derivations of this formula is completing square formula. It can be used in the following steps.

Firstly, you have to divide each side by a. Then rearrange. Next (b/2a)² has to be added to each side. Now, rearrange all the terms on the right side to get a common denominator. Then, square root both sides and finally isolate x.

4. What are Imaginary Roots to Quadratic Equation?

Ans. In algebra, you will often find that there are certain quadratic equations and their solution contain negative square root. These square roots are known as imaginary numbers, and the roots are called complex roots. However, these complex roots are expressed as a ± b.

Imaginary numbers in quadratic equations occur when the value of fundamental part of quadratic formula is in negative. In such cases, the equations do not contain zeros or roots in real number sets. 

In this equation, ax² + bx + c = 0, a, b and c are real numbers where a not = 0.