Lesson 7, Topic 1
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# Perfect Squares

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### 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

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.

Example 1

Write down the expansions of the following

a. (y + 8)2Â

= (y + 8)Â (y + 8)

= y (y + 8) + 8 (y + 8)

= y2 + 8y + 8y + 64

= y2 + 16y + 64

b. (y – 5)2Â

= (y – 5) (y – 5)

= y (y – 5) – 5 ( y – 5)

= y2 – 5y – 5y + 25

= y2 – 10y + 25

c. (7a -3)2Â

=Â  (7aÂ  -3)Â  (7a â€“ 3)

= 7a (7a – 3) – 3 (7a – 3)

= 49a2 â€“ 21a â€“ 21a + 9

= 49a2 â€“ 42a + 9

d. (2y â€“ 6)2Â

=Â  (2y â€“ 6) (2y â€“ 6)

= 2y (2y â€“ 6) â€“ 6 (2y â€“ 6)

= 4y2 – 12y – 12y + 36

= 4y2 â€“ 24y + 36

e. (5xÂ  + 4y)2Â

=Â  (5xÂ  + 4y) (5xÂ  + 4y)

= 5x (5x + 4y) Â  + 4y (5Â  + 4y)

= 25x 2 + 20xyÂ + 20xy + 16y2

= 25x2 + 40xy + 16y2

Example 2

Factorise the following expressions

i. x2 + 4x + 4

x2 + 2x + 2x  + 4

(x2 + 2x)Â + (2xÂ +Â 4)

x (x  + 2) + 2(x  + 2)

(xÂ + 2) (xÂ + 2)
= (xÂ + 2)2

ii. x2 + 8xÂ  + 16

x2 + 4x  + 4x  + 16

(x2 + 4x)  + (4x  + 16)

x(x + 4 )  + 4(x  +  4)

(xÂ + 4) (xÂ + 4)
=Â  ( x2 + 4)

iii. Â x2 – 6xÂ  + 9

Â x2 – 3xÂ  – 3xÂ  + 9

(x2 – 3x) – (3x  – 9)

x(x  – 3) – 3(x – 3)

= (xÂ  – 3)(xÂ  – 3)
= (xÂ  – 3)2

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