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 – \(\sqrt{12}\)

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