Perfect Squares

A perfect square with a + sign between the two numbers:

(x + a)²  =  ( x + a) (x  + a)

 ⇒ x (x + a) + a (x  + a)

 ⇒ x² + ax + ax  + a²

 ⇒ (x + a)²  = x² + 2ax  + a²

We can state the above result as follows:

The first term squared + twice their product + the square of the second term.

A perfect square with a negative sign between the two numbers:

(x – a)²  =  (x – a) (x – a)

 ⇒ x(x – a) – a (x – a)

 ⇒ x² – ax  – ax  – a²

⇒ x² – 2ax  – a²

⇒ (x – a)²  =   x² – 2ax + a².

We can state the above result as follows:

the first term squared – twice their product + the square of the second term.

Worked Example 7.1.1:

Write down the expansions of the following:

a. (y + 8)² 
b. (y – 5)²
c. (7a – 3)² 
d. (2y – 6)² 
e. (5x  + 4y)² 

Solution

a. (y + 8)² 

⇒ (y + 8) (y + 8)

⇒ y (y + 8) + 8 (y + 8)

⇒ y² + 8y + 8y + 64

= y² + 16y + 64

b. (y – 5)² 

⇒ (y – 5) (y – 5)

⇒ y (y – 5) – 5 ( y – 5)

⇒ y² – 5y – 5y + 25

= y² – 10y + 25

c. (7a – 3)² 

⇒  (7a  – 3)  (7a – 3)

⇒ 7a (7a – 3) – 3 (7a – 3)

⇒ 49a² – 21a – 21a + 9

= 49a² – 42a + 9

d. (2y – 6)² 

⇒  (2y – 6) (2y – 6)

⇒ 2y (2y – 6) – 6 (2y – 6)

⇒ 4y² – 12y – 12y + 36

= 4y² – 24y + 36

e. (5x  + 4y)² 

⇒  (5x  + 4y) (5x  + 4y)

⇒ 5x (5x + 4y)   + 4y (5  + 4y)

⇒ 25x ² + 20xy + 20xy + 16y²

= 25x² + 40xy + 16y²

Worked Example 7.1.2:

Factorize the following expressions:

i. x² + 4x + 4
ii. x² + 8x  + 16
iii.  x² – 6x  + 9

Solution

i. x² + 4x + 4

⇒ x² + 2x + 2x  + 4

⇒ (x² + 2x) + (2x + 4)

 ⇒ x (x  + 2) + 2(x  + 2)

⇒ (x + 2) (x + 2)
= (x + 2)²

ii. x² + 8x  + 16

 ⇒ x² + 4x  + 4x  + 16

 ⇒ (x² + 4x)  + (4x  + 16)

⇒ x(x + 4 )  + 4(x  +  4)

⇒ (x + 4) (x + 4)
=  ( x + 4)²

iii.  x² – 6x  + 9

⇒ x² – 3x  – 3x  + 9

⇒ (x² – 3x) – (3x  – 9)

⇒ x(x  – 3) – 3(x – 3)

⇒ (x  – 3)(x  – 3)
= (x  – 3)²