# How to Solve Quadratic Equations By Factoring (Method And Examples)

Are you on the lookout for an easy way to solve quadratic equations? Well, then here is a simple way to solve a quadratic equation and to find the roots of the given equation by factorization method.

Suganya Sukumar

Last Updated: Jun 3, 2018

**Definitions of a Quadratic Equation**

Quadratic Polynomial

A polynomial of a second degree is called a quadratic polynomial. The general form of a quadratic polynomial is

**ax**, where a, b, c are real numbers, a ≠ 0 and x is a variable.^{2}+ bx + cExample

x

^{2}+ 2x + 1; 3x^{2}+ √6xQuadratic Equation

An equation p(x) = 0, where p(x) is a quadratic polynomial, is called a quadratic equation. The general form of a quadratic equation is,

**ax**where a, b, c are real numbers, a ≠ 0 and x is a variable.^{2}+ bx + c = 0Example

x

^{2}- 6x + 2 = 0**Roots of a Quadratic Equation**

- Consider the general form of a quadratic equation i.e., ax
^{2}+ bx + c = 0. - Factorize the term 'ac' such that the sum of the factors is equal to b.

**Solving Problems by Factorization Method**

**(i)**

**x**

^{2}- 5x + 6 = 0Solution

a = 1, b = -5 and c = 6;

a × c = 1 × 6 = 6;

a × c = 1 × 6 = 6;

The factors of 6 whose sum is equal to -5 is -3 and -2;

-5x can be replaced with -3x and -2x;

-5x can be replaced with -3x and -2x;

x

x(x-3) - 2(x-3) = 0;

^{2}- 3x -2x + 6 = 0;x(x-3) - 2(x-3) = 0;

Now, collect the common term in the bracket (x-3) and make the equation in the following way,

(x-3)(x-2) = 0;

If the product of a and b is zero, i.e., ab = 0, then either a = 0 or b = 0;

∴ x-2 = 0 or x-3 = 0;

So, x = 2 or x = 3

**Thus, the roots of the quadratic equation x**^{2}- 5x + 6 = 0 are 3 and 2.**(ii)**

**2x**

^{2}+ x - 3 = 0Solution

a = 2, b = 1 and c = -3;

a × c = -6;

a × c = -6;

The factors of -6 whose sum is equal to 1 is 3 and -2;

2x

2x

2x (x-1) + 3 (x-1) = 0:

(2x+3)(x-1) = 0;

^{2}+ 3x -2x - 3 = 0;2x

^{2}- 2x + 3x - 3 = 0;2x (x-1) + 3 (x-1) = 0:

(2x+3)(x-1) = 0;

x = 1 or x = - 3/2

**Thus, the roots of the quadratic equation 2x**

^{2}+ x - 3 = 0 are 2 and -3/2.**(iii)**

**x (x + 7) = 0**

Solution

This is a simple problem but there is a common error which people make, by multiplying the bracket term (i.e) x+7 with the x and making it a quadratic equation and to factorize it. But the logic is that, it is already factored!

(x+0)(x+7) = 0;

∴ (x+0) = 0 or (x+7) = 0;

∴ (x+0) = 0 or (x+7) = 0;

Thus x = 0 or x = -7

**Thus, the roots of the quadratic equation x (x + 7) = 0 are 0 and -7.****(iv)**

**x**

^{2}- 25 = 0

__Solution__x

The quadratic equation is of the form a

^{2}- 5^{2}= 0;The quadratic equation is of the form a

^{2}- b^{2}= (a+b)(a-b);So, (x+5)(x-5) = 0;

∴ (x+5) = 0 or (x-5) = 0;

∴ (x+5) = 0 or (x-5) = 0;

**Thus, x = -5 or x = 5**

**How to Check the Quadratic Equation**

If you are a beginner, it's always better to check the results for your confirmation.

^{2}- 5x + 6 = 0;

*for x = 3*;

x

^{2}- 5x + 6 = 0;

L.H.S (Left Hand Side)= 3

^{2}- 5(3) + 6;

= 9 - 15 + 6;

= 0 = R.H.S (Right Hand Side) proved.

By doing this, you can be confident that your solution is right.