Solution of a Quadratic Equation by Completing the square

In the previous section, we have learnt method of factorisation for obtaining the roots of a quadratic equation.

Is method of factorization applicable to all types of quadratic equation?

Let us try to solve x2 + 4x - 4 = 0 by factorisation method.

To solve the given equation x2 + 4x - 4 = 0 by factorization method.

We have to find ‘p’ and ‘q’ such that

p + q = and p x q =

We have no integers p,q satisfying above equation. So by factorization method it is difficult to solve the given equation.

Therefore, we shall try another method.

Consider the following situation

The product of Sunita's age (in years) two years ago and her age four years hence is one more than twice her present age.

What is her present age?

To answer this, let her present age be x years. Their age two years ago would be (x-2) and the age after four years will be (x+4) So, the product of both the ages is ((x-2)(x+4))  

Therefore, (x-2)(x+4) = 2x + 1

i.e.,x2+ 2x- 8 = 2x + 1

i.e., x2 - 9 =

So, Sunita's present age satisfies the quadratic equation x2- 9 = 0

So, Sunita's present age is years.

Now consider another quadratic equation (x+22 - 9 = 0) To solve it, we can write it as

x+22 =

x + 2 =  

Therefore, x = x =

So, the roots of the equation (x+22 - 9 = 0) are 1 and -5.

In both the examples above, the term containing x is inside a square, and we found the roots easily by taking the square roots.

But, what happens if we are asked to solve the equation (x2 + 4x -4 = 0) which cannot be solved by factorisation also.

So, we now introduce the method of completing the square. The idea behind this method is to adjust the left side of the quadratic equation so that it becomes a perfect square of the first degree polynomial and the RHS without x term.

The process is as follows:  

x2 + 4x - 4 = 0

=> x2 + 4x =

x2 + 2.x.2 = 4

Now, the LHS is in the form of (a2+2ab).

If we add (b^{2}) it becomes as (a2+ 2ab + b2) which is perfect square.

So, by adding (b2 =22 = ) to both sides we get,

x2 + 2.x.2 + 22 = 4 +

=> x+22=

=> x + 2 = ±√{8}

x = -22 ± 2√2.

Algorithm : Let the quadratic equation be (ax2 + bx + c = 0) ((a ≠ 0))

Step-1: Divide each side by 'a'

Step-2: Rearrange the equation so that constant term ca is on the right side. (RHS)

Step-3: Add 12ba2 to both sides to make LHS, a perfect square.

Step-4: Write the LHS as a square and simplify the RHS.  

Step-5: Solve it.

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