Solution of a Quadratic Equation by Factorisation

In Mathematics, a quadratic equation of variable x is an equation, which is in the standard form ax2+bx+c = 0, where a, b and c are the numbers and the coefficient of x2 should not be equal to zero (i.e) a ≠ 0. In other words, a quadratic equation is an equation whose degree of a polynomial is equal to 2. The solution of a quadratic equation is called the roots of the quadratic equation, which is found using two different methods, such as the factorisation method and the quadratic equation formula method. In this article, we will learn how to find the solution of a quadratic equation by the factorisation method with many solved examples.

How to Find the Solution of a Quadratic Equation by Factorisation?

For a quadratic equation, a real number α is called the root of a quadratic equation ax2+bx+c =0. Hence, we can write aα2 + bα + c = 0. So, x= α is the solution of a quadratic equation or the root of a quadratic equation. In other words, α satisfies the given quadratic equation.

(Note: The zeros of the quadratic equation ax2+bx+c = 0 are the same as the roots of the quadratic equation ax2+bx+c = 0.)

Now, let us understand the procedure to find the solution of a quadratic equation with the help of examples.

Example 1:

Solve the quadratic equation 2x2+x-300 = 0 by the factorisation method.

Solution:

Given quadratic equation: 2x2+x-300 = 0

By using factorisation, the quadratic equation 2x2+x -300 = 0 is written as:

2x2 – 24x+25x -300 = 0

2x(x-12) +25(x-12) =0

(i.e) (x-12)(2x+25) = 0

Therefore, x-12=0 and 2x+25 = 0

x-12 = 0

Therefore, x= 12.

Similarly, 2x+25 = 0

2x= -25

x =-25/2

x = -12.5.

Hence, the roots of the quadratic equation 2x2+x-300 = 0 are 12 and -12.5.

Example 2:

Find the roots of the quadratic equation 6x2-x-2 =0 using the factorisation method.

Solution:

Given quadratic equation: 6x2-x-2 =0

By using the factorisation method, 6x2-x-2 =0 is written as 6x2+3x-4x-2 =0

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

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

Therefore, (2x+1) =0 and (3x-2)=0

Hence, 2x+1 = 0

2x = -1

x = -½.

Similarly, 3x-2=0

3x = 2

x = ⅔.

Therefore, the roots of the quadratic equation are -½ and ⅔.

Practice Problems

Solve the following problems.

  1. Find the roots of the quadratic equation 2x2+x-6 = 0, using the factorisation method.
  2. Find the zeros of the quadratic equation 100x2-20x+1=0 using factorisation.
  3. Solve the quadratic equation 2x2-x+(⅛) =0.

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