Completing a Square

Here we will learn how to completing a square.

a2x2 + bx = a2x2 + 2 ∙ ax ∙ b2a

               = {(ax)2 + 2ax ∙ b2a + (b2a)2} - (b2a)2

               = (ax + b2a)2 - (b2a)2


a2x2 - bx = a{x2 - bax}

              = a{x2 - 2x ∙ b2a + (b2a)2} - a ∙ (b2a)2

              = a(x - b2a)2 - b24a

Solved Examples on Completing a Square:

1. What should be added to the polynomial 4m2 + 8m so that it becomes perfect square?

Solution:

4m2 + 8m

= (2m)2 + 2 ∙ (2m) ∙ 2

= (2m)2 + 2 ∙ (2m) ∙ 2 + 22 – 22

= (2m + 2)2 – 4.

Therefore, (4m2 + 8m) + 4 = (2m + 2)2 – 4 + 4 = (2m + 2)2.

So, 4 is to be added to 4m2 + 8m to make it a perfect square.


2. What should be added to the polynomial 9k2 – 4k so that it becomes perfect square?

Solution:

9k2 – 4k

= (3k)2 - 2 ∙ (3k) ∙ 23

= (3k)2 - 2 ∙ (3k) ∙ 23 + (23)2 - (23)2

= (3x - 23)2 – 49

Therefore, (9k2 – 4k) + 49 = (3x - 23)2 – 49 + 49  = (3x - 23)2

So, 49 is to be added to 9k2 - 4k to make it a perfect square.


3. What should be added to 16m4 + 9 to make it a whole square of a polynomial of the second degree?

Solution:

16m4 + 9

= (4m2)2 + 32

= (4m2)2 ± 2 ∙ (4m2) ∙ 3 + 32 ∓ 2 ∙ (4m2) ∙ 3

= (4m2 ± 3)2 ∓ 24m2

Therefore, (16m4 + 9) ± 24m2 = (4m2 ± 3)2 ∓ 24m2 ± 24m2

= (4m2 ± 3)2.

So, ± 24m2 is to be added to 16m4 + 9 to make it s whole square of a polynomial of the second degree.


4. Find k so that p2 – 5p + k can be a perfect square of a linear polynomial.

Solution:

p2 – 5p + k

= p2 – 2p ∙ 52 + k

= p2 – 2p ∙ 52 + (52)2 + k - (52)2

= (p - 52)2 + (k - 254)

So, p2 – 5p + k can be perfect square if k - 254 = 0, i.e., k = 254.






9th Grade Math

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