Solve Quadratic Equation by Completing the Square

Here, you will learn completion of square method and how to solve quadratic equation by completing the square with examples.

Let’s begin – 

Solve Quadratic Equation by Completing the Square

Let us consider the equation $x^2 + 8x + 4$ = 0

If we want to factorize the left hand side of the equation using the method of splitting the middle term, we must determine two integer factors of 4 whose sum is 8.

But the factors of 4 are 1, 4; -1, -4; -2, -2; and 2, 2. In these cases the sum is not 8.

Therefore, using factorization, we cannot solve the given equation $x^2 + 8x + 4$ = 0.

Here, we shall discuss a method known as completing the square to solve such quadratic equations.

In the method completion of square we simply add and subtract $({1\over 2} coefficient of x)^2$ in LHS. 

Let’s understand the concept of completing the square by taking an example.

Example : Solve the given quadratic equation $x^2 + 8x + 4$ = 0 by using completion of square method.

Solution : We have, $x^2 + 8x + 4$ = 0

We add and subtract $({1\over 2} coefficient of x)^2$ in LHS and get

$x^2 + 8x + ({1\over 2}\times 8)^2 – ({1\over 2}\times 8)^2$ + 4 = 0

$x^2 + 8x + 16 – 16 + 4$ = 0

$x^2 + 2(4x) + (4)^2 – 12$ = 0

$(x + 4)^2$ – $(\sqrt{12})^2$ = 0

$(x + 4)^2$ = $(\sqrt{12})^2$

$\implies$ x + 4 = $\pm \sqrt{12}$

$\implies$ x = -4 $\pm \sqrt{12}$

This gives  x = -4 + $\sqrt{12}$   or    x = -4 – $\sqrt{12}$