Factoring (Part 3)

Special Cases and

Common Factoring

On the 9th day of quadratics

my teacher gave to me…

Learning Goals

By the end of the lesson I will be able to:

• Factor out a value common to all terms in a quadratic relation
• Factor the special case called a difference of squares

Review and Warm up

Using the Sum-Product method, when we factor a quadratic like x² + 7x - 18, we first look for 2 integers that add up to +7 (the b value) and multiply to -18 (the c value.)

What if you were asked to factor:

a) x² + 3x b) x² - 12x

c) x² - 9 d) x² - 36

Common Factoring

The first two cases can be solved quickly using a technique called “common factoring.”

Common factoring is when you go looking for the common elements (in most cases x and/or an integer) in every term and pull just those elements out to be multiplied by what is left.

In Painful detail

If we have a quadratic relation:

y = x² - 12x

We could rewrite this as:

y = xx - 12x

Is there something common in both terms?

An x! We can factor out a single x from both terms.

y = x(x - 12)

Explained differently

We are essentially dividing every term by x:

y = x² - 12x

x x

= x(x - 12)

What are the x-intercepts?

(x + 0)(x - 12) → x-intercepts of 0 and 12

Rapid Fire!

Factor the following, and determine the x-intercepts

y = x² + 3x

= x(x + 3)

x-int: 0, -3

y = x² - 5x

= x(x - 5)

x-int: 0, 5

y = x² + 4x

= x(x + 4)

x-int: 0, -4

y = x² - 2x

= x(x - 2)

x-int: 0, 2

y = x² - 8x

= x(x - 8)

x-int: 0, 8

y = x² - 15x

= x(x - 15)

x-int: 0, 15

y = x² + 42x

= x(x + 42)

x-int: 0, -42

y = x² - 99x

= x(x - 99)

x-int: 0, 99

Difference of Squares

Remember back to the warm up? The other examples:

c) x² - 9 d) x² - 36

There’s a trick to factoring those quickly too!

But let’s talk about what we’re actually looking for first...

Faster!

y = x² - 9

Find 2 numbers that:

• multiply to -9
• add to 0

So our r and s values have to be the same:

r = s

That means that r = s = 9 = +3 and -3

y = x² - 9 = (x + 3)(x - 3)

Notes

In order to use the difference of squares you have to be factoring a relation that has:

• A “c” value that is negative
• A “b” value that is 0 (there is no x term)

To find the 2 factors take the square root of the c value (without the sign)

Rapid Fire!

Factor the following, and determine the x-intercepts

y = x² - 4

= (x - 2)(x + 2)

x-int: +2, -2

y = x² - 16

= (x - 4)(x + 4)

x-int: +4, -4

y = x² - 25

= (x - 5)(x + 5)

x-int: +5, -5

y = x² - 49

= (x - 7)(x + 7)

x-int: +7, -7

y = x² - 36

= (x - 6)(x + 6)

x-int: +6, -6

y = x² - 121

= (x - 11)(x + 11)

x-int: +11, -11

Factoring when A is not 1

So far we have dealt only with quadratics where the a value is 1: y = x² + 2x + 1, y = x² - 9, y = x² + 6x

What if the coefficient “a” isn’t 1? Can we factor that?

y = 3x² - 9x + 6

Not yet, but if we factor out a 3 from all 3 terms:

y = 3(x² - 3x + 2)

= 3(x - 1)(x - 2)

Step by step

Factor y = 5x² + 5x - 30

1. Factor out the a value (5)

y = 5(x² + x - 6)

2. Factor what is inside the brackets normally:

adds to 1, multiplies to -6: +3 and -2

y = 5(x + 3)(x - 2)

Speed Round!

Factor the following and identify the x-intercepts:

y = -2x² - 2x + 24

= -2(x² + x - 12)

= -2(x + 4)(x - 3)

x-int: -4, +3

y = 3x² - 24x + 48

= 3(x² - 8x + 16)

= 3(x - 4)(x - 4)

x-int: 4

y = -x² + 5x + 14

= -(x² - 5x - 14)

= -(x + 2)(x - 7)

x-int: -2, +7

y = -8x² - 16x - 8

= -8(x² + 2x + 1)

= -8(x + 1)(x + 1)

x-int: -1

All together now

Factor the standard form equation below, then graph it.

y = -2x² - 6x + 8