Solution of a Quadratic Equation by Completion of Squares

If we have to solve any quadratic equation, we would probably switch to factorization method. But in some cases, it is not convenient to solve quadratic equations by factorization method.

For example, consider the equation x² + 4x + 2 = 0. In order to solve this equation by factorization method we will have to split the coefficient of the middle term 4 into two integers whose sum is 4 and product is 2. Clearly, this is not possible in integers. Therefore, we cannot solve this equation by using factorization method.

We can convert any quadratic equation to the form (x + a)² – b² = 0 and can easily find its roots. Let us consider the equation 3x² – 5x + 2 = 0. Here coefficient of x² is not a perfect square. So, we multiply the equation throughout by 3 to get 9x² – 15x + 6 = 0

Now we have, 9x² – 15x + 6

= (3x)² – 2 x (3x) x 5/2 + 6

= (3x)² – 2 x (3x) x 5/2  + (5/2)² – (5/2)² + 6

= (3x-5/2)² – 25/4 + 6

= (3x-5/2)² – ¼

(3x-5/2)² = ¼

3x-5/2 = 1/2 or 3x-5/2 = -1/2

3x=1/2+5/2 or 3x=-1/2+5/2

3x=6/2 or 3x = -4/2

x=3/3 or x=-2/3

x=1 or x = -2/3

Here is a formula for finding the roots of any quadratic. It is proved by completing the square. In other words, the quadratic formula completes the square for us. This method is popularly known as Sridharacharya’s formula as it was first given by an ancient Indian mathematics Sridharacharya around 1025 A.D

Quadratic Formula

Thus, if b² – 4ac ≥ 0, then the quadratic equation ax² + bx + c = 0 has two roots α and β given by

Nature of the Roots of a Quadratic Equation

It is clear that the roots of the quadratic equation ax² + bx + c = 0, a ≠ 0 are real if b² – 4ac ≥ 0 and if b² – 4ac < 0, the equation will have no real roots. In case b² – 4ac = 0 then the two roots are equal or coincident each being equal to –b/2a.It is clear that the nature of the roots of the quadratic equation ax² + bx + c = 0, a ≠ 0 depends upon the value of the expression b² – 4ac. This expression is generally called the discriminant and is denoted by D.

Points To Be Remembered

Let ax² + bx + c = 0, a ≠ 0 be a quadratic equation, then D = b² – 4ac is called the discriminant of the equation.

1   Example 1: Determine whether the given quadratic equations have real roots and if so, find the roots. 9x² + 7x – 2 = 0

Solution: The given equation is 9x² + 7x – 2 = 0.

Here, a = 9, b = 7 and c = -2.

Therefore D = b² – 4ac = 7² – 4x9x-2 = 49 + 72 = 121 > 0

So, the given equation has real roots, given by

Example 2: Find the value of k for which the given equation has real and equal roots.

2x² – 10x + k = 0

Solution: The given equation is 2x² – 10x + k = 0

Here, a =2, b=-10 and c=k

Therefore, D = b² – 4ac = (-10)² – 4 x 2 x k = 100 – 8k

The given equation will have real and equal roots if D = 0

100 – 8k = 0

k = 100/8 = 25/2

Example 3: If -4 is a root of the quadratic equation x² + px – 4 = 0 and the quadratic equation x² + px + k = 0 has equal roots, find the value of k.

Solution: Since -4 is a root of the equation x² + px – 4 = 0

Therefore (-4)² + p x -4 – 4 = 0 (As root always satisfies the equation)

16 – 4p – 4 = 0

4p = 12

P = 12/4 = 3

The equation x² + px + k = 0 has equal roots.

Therefore Discriminant = 0

P² – 4k = 0

9 – 4k = 0

K = 9/4