Are any of these perfect squares?
No, these are perfect
squares

Recall: A perfect square is a number that is
obtained by a product of the same number.
◦ EX. 16 and 25 are perfect squares, because
4 x 4 = 16 and 5 x 5 = 25
 22 on the other hand is not a perfect square because
11 x 2 = 22

Now let’s look at what it means to be a
“perfect square” in the context of a
quadratic equation

We agree that a number multiplied by itself
will return a perfect square
◦ (5) x (5) = (5)2 = 25

This is true for anything in the brackets
◦ EX. (☺) x (☺) = (☺)2
◦ EX. (♥) x (♥) = (♥)2

So how do you think it is possible to state
that the equation y = 4x2 + 12x + 9 is a
perfect square?



y = 4x2 + 12x + 9
Based on our previous conclusion, if we can
write the expression as (something)2, it is a
perfect square
But what times itself gives 4x2 + 12x + 9?

In y = 4x2 + 12x + 9, both the first and last
numbers (4 and 9) are perfect squares
◦ 22 = 4 and 32 = 9 – we can use this


It turns out, that 4x2 + 12x + 9 = (2x + 3)2
This trick usually works, but expand the
brackets to verify that


Factor 25x2 – 40x + 16
Using the trick we just found:
◦ 52 = 25, and 42 = 16, but here, the middle term is
negative, so perhaps it is (5x – 4)2

If you check this, you will see that it is correct
◦ (5x – 4)(5x – 4)
 = 25x2 – 20x – 20x + 16
 = 25x2 – 40x + 16

Factor the following difference of squares:
◦ x2 – 1
◦ The coefficient in front of the x2 is 1
 1 x 1 = (1)2 = 1
 x2 – 1 = (x + 1)(x – 1)
◦ So instead of being just equal to (x + 1)2
or (x – 1)2, it is equal to (x + 1)(x – 1) –
this ensures that the middle term
(with a single x) cancels out.
Factor: x2 – 64
𝑥 2 = 𝑥; 64 = 8
x2 – 64 = (x+8)(x-8)

Check by expanding:
(x+8)(x-8)
= x2 – 8x + 8x – 64
= x2 - 64

Factor: 81x4 – 25y2
81𝑥 4 = 9𝑥 2 ;
25𝑦 2 = 5𝑦
81x4 – 25y2
= (9𝑥 2 +5𝑦)(9𝑥 2 − 5𝑦)
Check by expanding:
(9𝑥 2 +5𝑦)(9𝑥 2 − 5𝑦)
= 81x4 – 45x2y + 45x2y –
25y2
=81x4 – 25y2

A polynomial of the form a2 + 2ab + b2 or
a2 – 2ab + b2 is a perfect square trinomial:
◦ a2 + 2ab + b2 can be factored as (a + b)2
◦ a2 - 2ab + b2 can be factored as (a - b)2

A polynomial of the form a2 – b2 is a
difference of squares and can be factored as
(a + b)(a – b)