Completing the Square Formula Examples

Example 1: Find the roots of the quadratic equation of the x² + 2x – 12 = 0 by using the method of completing the square.

Solution:

Given Quadratic equation is  x² + 2x – 12 = 0

So as comparing the equation along with standard form,

where b =  2, and c = -12

Then (x + b/2)² = -(c – b²/4)

By substituting the values we get 

(x + 2/2)² = -(-12 – (2²/4) )

(x + 1)²  = 12 + 1

(x + 1)² = 13

x + 1 =  ± √13

x + 1 = ± 3.6

So, x + 1 = +3.6 and x+1 = – 3.6

x = 2.6 , -4.6

Therefore roots for the given equation are 2.6, -4.6.

Example 2: Find the roots of the quadratic equation of the 2x² – 4x – 20 = 0 by using the method of completing the square.

Solution:

Given Quadratic equation is 2x² – 4x – 20 = 0

Supplied equation is not in the form to which the method of completing squares is used, i.e. the x2 coefficient is not 1. To make it one, divide the entire equation by  2 .

then x² – 2x – 10 = 0

So as comparing the equation along with standard form,

where b = – 2, and c = -10

then (x + b/2)² = -(c – b²/4)

by substituting the values we get

(x + (-2/2))² = -( -10 – (2²/4))

(x – 1)²  =  11

x – 1 = ± √11

x – 1  = ± 3.3

So,  x – 1 = + 3.3 and x -1 = -3.3

x = 4.3,  -2.3

Therefore roots for the given equation are 4.3, -2.3.

Example 3: Solve Using the completing the square formula for 3x² – 9x – 27 = 0.

Solution:

Given Quadratic equation is 3x² – 9x – 27 = 0.

we can write it as x² – 3x -9 =0 

So as comparing the equation along with standard form,

where b = – 3, and c = -9

then (x + b/2)² = -(c – b²/4)

by substituting the values we get

(x + (-3/2) )² = -( -9 – (3²/4) )

(x – 1.5 )²  = 11.25

x – 1.5 = ± √11.25

x – 1.5  = ± 3.35

So,  x – 1.5 = + 11.25 and x -1 = -11.25

x = 12.75, -10.25

Therefore roots for the given equation are 12.75, -10.25.

Example 4: Find the number that needs be added to x² – 4x to make it a perfect square trinomial using the completing the square formula.

Solution:

Given expression is x²– 4x

As Comparing the given expression along with ax² + bx + c, 

a = 1; b = -4

Term that should be added to make the above expression a perfect square trinomial using the formula is,

(b/2a)² =  (-4/2(1))²       

(b/2a)² = 4

Therefore the number that needs be added to x² – 4x to make it a perfect square trinomial is 4.

Example 5: Find the number that needs be added to x² + 22x to make it a perfect square trinomial using the completing the square formula.

Solution:

Given expression is x² + 22x

As Comparing the given expression along with ax² + bx + c,

a = 1 ; b = 22

The term that should be added to make the above expression a perfect square trinomial using the formula is,

(b/2a)² =  ( 22/2(1) )²       

(b/2a)²  = 121

Therefore the number that needs to be added to x² + 22x to make it a perfect square trinomial is 121.

Completing the square is a method used to solve quadratic equations and to rewrite quadratic expressions in a different form. It helps us to find the solutions of the equation and to understand the properties of a quadratic function, such as its vertex.

In this article, we will learn about, Completing the Square Methods, Completing the Square Formula, Completing the Square Examples and others in detail.