Completing a Square is another method for the Expansion of Powers of Binomials and Trinomials. In the previous article, we have discussed different methods to expand and simplify binomial and trinomial expressions. The derivations of completing a square in expanding binomial expressions can be obtained here. On this page, we will learn deeply about completing a square with an explanation here.
What does Completing a Square Mean?
Completing Square Methos is one of the techniques to find Quadratic Equation Roots. We need to convert the given equation into a perfect square. We can determine the quadratic equation roots by using the quadratic formula.
How to Complete a Square?
a²x² + bx = a²x² + 2 . ax . b/2a
= {(ax)² + 2ax . b/2a + (b/2a)²} – (b/2a)²
= (ax + b/2a)² – (b/2a)²
a²x² – bx = a {x² – b/a . x}
= a {x² – 2x . b/2a + (b/2a)²} – a . (b/2a)²
= a (x – b/2a)² – (b/2a)²
Examples on Completing a Square
Check out the below examples to know more about the concept of completing a square.
Example 1.
What should be added to the polynomial 4x² + 16x so that it becomes a perfect square?
Solution:
Given the binomial expression 4x² + 16x
We have to convert the given expression into a perfect square.
We can write 4x² + 16x as
(2x)² + 2 (2x)(4)
Here a = 2x and b = 4
(2x)² + 2 (2x)(4) + 4² – 4²
(2x)² + 2 (2x)(4) + 16 – 16
(2x + 4)² – 4²
Thus we have to add 16 to make the given expression a perfect square.
Example 2.
What should be added to the polynomial x² + 2x so that it becomes a perfect square?
Solution:
Given the binomial expression x² + 2x
We have to convert the given expression into a perfect square.
We can write x² + 2x as
(x)² + 1 (2x)(1)
Here a = x and b = 1
(x)² + 1 (2x)(1) + 1² – 1²
(x)² + 1 (2x)(1) + 1 – 1
(2x + 1)² – 1²
Thus we have to add 1 to make the given expression a perfect square.
Example 3.
What should be added to the polynomial 9x² – 12x so that it becomes a perfect square?
Solution:
Given the binomial expression 9x² – 12x
We have to convert the given expression into a perfect square.
We can write 9x² – 12x as
(3x)² – 2 (3x)(2)
Here a = 3x and b = 2
(3x)² – 2 (3x)(2) + 2² – 2²
(3x)² – 2 (3x)(2) + 4 – 4
(3x – 2)² – 2²
Thus we have to add 4 to make the given expression a perfect square.
Example 4.
What should be added to the polynomial 4m² + 8m so that it becomes a perfect square?
Solution:
Given the binomial expression 4m² + 8m
We have to convert the given expression into a perfect square.
We can write 4m² + 8m as
(2m)² + 2 (2m)(2)
Here a = 2m and b = 2
(2m)² + 2 (2m)(2) + 2² – 2²
(2m)² + 2 (2m)(2) + 4 – 4
(2m + 2)² – 2²
Thus we have to add 4 to make the given expression a perfect square.
Example 5.
What should be added to the polynomial 16m² + 8m so that it becomes a perfect square?
Solution:
Given the binomial expression 16m² + 24m
We have to convert the given expression into a perfect square.
We can write 4m² + 8m as
(4m)² + 2 (4m)(3)
Here a = 4m and b = 3
(4m)² + 2 (4m)(3) + 3² – 3²
(4m)² + 2 (4m)(3) + 9 – 9
(4m + 3)² – 3²
Thus we have to add 9 to make the given expression a perfect square.