Here, you will learn completion of square method and how to solve quadratic equation by completing the square with examples.
Letโs begin โย
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.ย
Letโs 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
\(x^2 + 8x + ({1\over 2}\times 8)^2 โ ({1\over 2}\times 8)^2\) + 4 = 0
\(x^2 + 8x + 16 โ 16 + 4\) = 0
\(x^2 + 2(4x) + (4)^2 โ 12\) = 0
\((x + 4)^2\) โ \((\sqrt{12})^2\) = 0
\((x + 4)^2\) = \((\sqrt{12})^2\)
\(\implies\) x + 4 = \(\pm \sqrt{12}\)
\(\implies\) x = -4 \(\pm \sqrt{12}\)
This givesย x = -4 + \(\sqrt{12}\)ย ย orย ย x = -4 โ \(\sqrt{12}\)
