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 – $$\sqrt{12}$$