# Solving Quadratic Equations by Factorisation

The process of writing an expression as a product of two or more common factors is called the method of factorization. For example:

1. ${x^2} + 5x + 6 = \left( {x + 2} \right)\left( {x + 3} \right)$
2. $5{x^2} + 8x = x\left( {5x + 8} \right)$
3. $30 = 2 \times 3 \times 5$

In the above examples, $\left( {x + 2} \right)\left( {x + 3} \right)$ are the factors of expression ${x^2} + 5x + 6$, $x\left( {5x + 8} \right)$ are the factors of $5{x^2} + 8x$ and $2 \times 3 \times 5$ are the factors of $30$.

While solving the quadratic equation by the method of factorization, we have the following steps:

• Convert the quadratic equation into standard form if necessary, i.e. $a{x^2} + bx + c = 0$,    where $a \ne 0$.
• Multiply the coefficient of ${x^2}$ by constant terms, and we get $a \times c = ac$.
• Now try to find two numbers whose product is $ac$ and sum or difference is equal to $b$ (coefficient of $x$).
• Factorise the given expression on L.H.S.
• Equate each factor equal to zero.
• We get the required roots, say ${x_1}$, ${x_2}$.

Example:

Solve the equation by factorization method.

Solution:

The given equation in standard form is $5{x^2} - 11x + 6 = 0$

Multiplying the coefficient of ${x^2}$ and the constant term, we get $5 \times 6 = 30$

Divide $30$ into two parts such that their difference or sum is $11$

 Possible factors of $30$ Sum or difference of factors $30 \times 1 = 30$ $30 - 1 = 29,{\text{ }}30 + 1 = 31$ (not possible) $15 \times 2 = 30$ $15 - 2 = 13,{\text{ 15}} + 2 = 17$ (not possible) $10 \times 3 = 30$ $10 - 3 = 7,{\text{ 1}}0 + 3 = 13$    (not possible) $6 \times 5 = 30$ $6 - 5 = 1,{\text{ 6}} + 5 = 11$          (possible)

Therefore, $5{x^2} - 11x + 6 = 0$
$5{x^2} - \left( {5 + 6} \right)x + 6 = 0$
$5{x^2} - 5x - 6x + 6 = 0$
$5x\left( {x - 1} \right) - 6\left( {x - 1} \right) = 0$
$\left( {x - 1} \right)\left( {5x - 6} \right) = 0$

Either $x - 1 = 0$ or $5x - 6 = 0$
$x = 1$ or $x = \frac{6}{5}$

Example:

Solve the equation by factorization method.

Solution:

The given equation in standard form is $4{x^2} + 8x = 0$

Here, the constant term is absent; its factorization is very simple.

Taking common $4x$, we get
$4x\left( {x + 2} \right) = 0$

Either $4x = 0$ or $x + 2 = 0$
$x = 0$ or $x = - 2$