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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