4.5 Factoring Quadratics: Special Cases

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

Perfect Squares

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?

Perfect Squares

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?

Algebraic Perfect Squares

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

A Bit of Trial & Error…

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

Example #2

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.

Example #3

Factor: x2 – 64

x2 – 64 = (x+8)(x-8)

Check by expanding:(x+8)(x-8) = x2 – 8x + 8x – 64= x2 - 64

Example #4

Factor: 81x4 – 25y2

81x4 – 25y2

= (+)()Check by expanding:(+)()= 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)

In Summary…