Special Cases & Completing the Square

Difference of Perfect Squares Perfect Square Trinomial

Completing the Square

Dr. Tom Giles

Difference of Two Squares

Multiple Choice

Factor the following expression:

$$x^2-81$$  

• (x-9)(x-9)

• (x+9)(x+9)

• (x+9)(x-9)

• Not Factorable

Multiple Choice

Factor the following expression:

$$x^2-144$$  

• (x-12)(x+12)

• (x-12)(x-12)

• (x+12)(x+12)

• Not Factorable

Multiple Choice

Factor the following expression:

$$x^2+64$$  

• (x+8)(x+8)

• (x-8)(x-8)

• (x-8)(x+8)

• Not Factorable

Multiple Choice

Multiply the following factors:  $$\left(x+2\right)\left(x-2\right)$$  

• $$x^2+4x+4$$

• $$x^2-4x+4$$  

• $$x^{2\ }-4x-4$$  

• $$x^2-4$$  

• $$x^2+4x-4$$  

Multiple Choice

Multiply the following factors:  $$\left(x-3\right)\left(x+3\right)$$  

• $$x^2+9x+9$$

• $$x^2-9$$  

• $$x^{2\ }-9x-9$$  

• $$x^2-9x+9$$  

• $$x^2+9x-9$$  

Multiple Choice

Multiply the following factors:  $$\left(x-10\right)\left(x+10\right)$$  

• $$x^2-100$$

• $$x^2-50x+100$$  

• $$x^{2\ }-x-100$$  

• $$x^2-20x+20$$  

• $$x^2+x-10$$  

Multiple Select

Check all the expressions that are difference of two squares.

• $$x^2-25$$

• $$x^2-100$$

• $$x^2+36$$

• $$x^2-73$$

• $$16x^2-400$$

Open Ended

Why is

$$x^2+36$$ not difference of two squares? 

Open Ended

Why is

$$x^2-73$$ not difference of two squares? 

Poll

Is

$$x^2+49$$  factorable, by difference of two squares?

• Yes

• No

Perfect Square Trinomials

What is a Perfect Square Trinomial?

Multiple Choice

Can the expression $$\left(x+2\right)\left(x+2\right)$$

also be written as  $$\left(x+2\right)^2$$ ?

• Yes

• No

Multiple Choice

Expand the following expression: $$\left(x+2\right)^2$$

.

• $$x^2+4x+4$$  

• $$x^2-4x+4$$  

• $$x^2-4$$  

• $$x^2+4$$  

Multiple Choice

Expand the following expression: $$\left(x-2\right)^2$$

.

• $$x^2+4x+4$$  

• $$x^2-4x+4$$  

• $$x^2-4$$  

• $$x^2+4$$  

Multiple Choice

Expand the following expression: $$\left(x+6\right)^2$$

.

• $$x^2+12x+36$$  

• $$x^2-12x+36$$  

• $$x^2-36$$  

• $$x^2+36$$  

Multiple Choice

Expand the following expression: $$\left(x-6\right)^2$$

.

• $$x^2+12x+36$$  

• $$x^2-12x+36$$  

• $$x^2-36$$  

• $$x^2+36$$  

Multiple Select

Select all that are factors of  the following expression: $$x^2+16x+64$$

.

• $$\left(x+4\right)\left(x+4\right)$$  

• $$\left(x+8\right)^2$$  

• $$\left(x+8\right)\left(x+8\right)$$  

• $$\left(x+16\right)^2$$  

Multiple Select

Select all that are factors of  the following expression: $$x^2-20x+100$$

.

• $$\left(x-10\right)\left(x-10\right)$$  

• $$\left(x+5\right)\left(x-20\right)$$  

• $$\left(x+20\right)\left(x-5\right)$$  

• $$\left(x-10\right)^2$$  

Multiple Select

Select all that are factors of  the following expression: $$x^2-6x+9$$

.

• $$\left(x-4.5\right)\left(x-2\right)$$  

• $$\left(x-3\right)\left(x-3\right)$$  

• $$\left(x-9\right)\left(x-1\right)$$  

• $$\left(x-3\right)^2$$  

Multiple Select

Select all that are factors of  the following expression: $$x^2+10x+25$$

.

• $$\left(x+5\right)\left(x+5\right)$$  

• $$\left(x+10\right)\left(x+15\right)$$  

• $$\left(x+5\right)^2$$  

• $$\left(x-5\right)^2$$  

Multiple Select

Select all that are factors of  the following expression: $$x^2+26x+169$$

.

• $$\left(x+17\right)\left(x+17\right)$$  

• $$\left(x+13\right)\left(x+13\right)$$  

• $$\left(x+13\right)^2$$  

• $$\left(x+17\right)^2$$  

Completing the Square

How do I complete the square?

Completing the Square

How do I complete the square?

Let's Practice!

Multiple Choice

Find the last term:

$$x^2+22x+c$$  

• 22

• 11

• 121

• 44

Multiple Choice

Find the last term:

$$x^2-8x+c$$  

• 8

• 4

• -16

• 16

Multiple Choice

Find the last term:

$$x^2-16x+c$$  

• 8

• 64

• -16

• 16

Multiple Choice

Find the last term:

$$x^2+34x+c$$  

• 289

• 17

• 34

• -34

Multiple Choice

Find the last term:

$$x^2+30x+c$$  

• 30

• 15

• 225

• -15

Multiple Choice

Find the last term:

$$x^2+4x+c$$  

• 4

• 2

• -2

• -4

Multiple Choice

Find the last term:

$$x^2-6x+c$$  

• -9

• 9

• 6

• -6

Multiple Choice

Find the last term:

$$x^2+10x+c$$  

• 5

• 2

• 20

• 25

Multiple Choice

Find the last term:

$$x^2+12x+c$$  

• 36

• 21

• 16

• 25

Multiple Choice

Find the last term:

$$x^2+42x+c$$  

• -9

• 21

• 441

• 1764

Multiple Choice

Find the last term:

$$x^2-2x+c$$  

• -2

• 1

• -1

• 2

Multiple Choice

Find the last term:

$$x^2-28x+c$$  

• 14

• 200

• 169

• 196