CRASH COURSE IN QUADRATICS

In preparation for the Algebra CST

-b + b2 – 4

ac

2ac

√

(x+4)(x-3)=0

(x+1)(x+2)

X2 – 5x +4

F O I L

CompleteThe Square

Multiplying Polynomials

Area Model of Multiplication

(30)(60)1800

(30)(8)240

(4)(60)240

(4)(8)32

60 + 8

30

+

4

1800+240+240+32=2312

To multiply 68 x 34:• Write the two numbers in

expanded notation and multiply one box at a time.

• After you have multiplied the numbers, add all of the products together.

Now you try one… 48 x 53

Multiplying Polynomials

Area Model of Multiplication

(x)(x)x2

(x)(2)2x

(3)(x)3x

(3)(2)6

x + 2

x

+

3

X2 + + 6

To multiply (x+2)(x+3):• Write the two numbers in

expanded notation and multiply one box at a time.

• After you have multiplied the numbers, add all of the products together. 5x

Now you try one… (x+5)(x+1)

Multiplying Polynomials

FOIL ( x + 2 ) ( x + 3)

First (x)(x) = x2

Outer (x)(3) = 3x

Inner (2)(x) = 2x

Last (2)(3) = 6

Combine like terms…

= x2 + 5x + 6

Multiplying Polynomials

x2 + 5x + 6 ax2 + bx + c

a = 1b = 5c = 6

Factoring Polynomials

3 4

12

7

2 5

10

7

6 1

6

7

7 2

14

9

3 5 6 4

18

9

21

10

Ask yourself… “What two numbers multiplied together give you the top digit and added together give you the bottom?”

Factoring Polynomials

X2 + 7x + 127

12 (x + )(x+ )

X2 + 13x + 3613

36 (x + )(x+ )

(x + )(x+ )-6

-40X2 - 6x - 40

Perfect Square Trinomial

X2 + 12 + 36

X * X 6 * 6

(x + 6)(x + 6) (x + 6)2

X2 - 14 + 49

X * X 7 * 7

(x - 7)(x - 7) (x - 7)2-

Solving Quadratic Equations

• Graphing

• Factoring

• Using Square Roots

• Completing the Square

• Quadratic Formula

Graphing Quadratic Equations

x2 – 4x = 0

x y=x2 - 4x

y x, y

0 02 – 4(0) 0 0, 0

2 22 - 4(2) -4 2, -4

4 42 – 4(4) 0 4, 0

The Solution is the ________________

Find the solution for each graph:

Factoring Quadratic Equations

Using the Zero Product Property

(x-3)(x+7)=0

(x-3)=0 (x+7)=0

x = 3 x = -7

Factoring Quadratic Equations

Solve using the Zero Product Property

(x-3)(x+4)=0

(x+3)(2x-8)=0

(3x-1)(4x+1)=0

(3x+1)(8x-2)=0

Can you solve in your head?(x-2)(x+1)=0

x2 + 12x + 36

x =

x2 - 21x = 72

x =

-72

-21

If x2 is added to x, the sum is 42. What are the values of x?

Using Square Roots

Square-Root Property

x2 = 16

√x2 = √16

4x2 – 25 = 0

x = +4

+25 +25

√4x2 = √25

2x = 52 2

x = + 2.5

4x2 = 25

x2 = 16(4)2= 16

(-4)2 = 16

Completing the Square

Using Algebra Tiles x2 + 6x a= 1 b=6 c=0

b2( )

2 ( )62

2

x2 + 6x = 0 b = 6+ 9 + 9

x2 + 6x + 9 = 9

(x+3)(x+3)=9

(x+3)2 = 9

√(x+3)2 = √ 9

x+3 = 3x+3= 3

+x+3= -3

x = 0

x = -6

( )62

2= 9

Completing the Square

x2 + 14x = 15 b = 14 14 2( )

2

= 72 =49

Add to both sides of the

equation

+ 49 + 49

x2 + 14x + 49 = 64

Factor the

Perfect Square

(x+7)(x+7)=64

(x+7)2 = 64

√(x+7)2 = √ 64

x+7 = 8x+7= 8

+x+7= -8

x = 1

x = -15

Completing the Square

x2 - 10x = -1 3

b = 10 3 Add to

both sides of

the equation

Factor the

Perfect Square

3x2 – 10x = -33 3 3

-10 1 3 2

( )2

* = 100 36

Reduce

25 9

x2 - 10x = -1 3

+25 9

+25 9 -9 + 25

9 9=16

9x2 - 10x + 25 = 16 3 9 9

x – 5 3( ) 4

3=+x – 5

3√( )2

16 9

=√

x – 5 = 4 3 3x – 5 = -4 3 3

x = 9 3

x = 1 3

x – 5 3( ) 16

9=2

Completing the Square

x - 8x = 12

x - 8x = 5 What should be added to both sides of this equation?

x + 4x = 6

x - 4x = 8

ax – bx = c

2

2

2

2

2

The Quadratic Formula

x2 + 5x + 6 ax2 + bx + ca = 1b = 5c = 62x2 + 3x – 5 = 0

ax2 + bx + ca = 2 b = 3 c = -5

-b + √ b2 – 4ac

2ax =

-b + √ b2 – 4ac

2ax =

-3 + √ 32 – 4(2)(-5)

2(2)x =

-3 + √ 9 – (-40)

4x = -3 + √ 49

4x =

-3 + 7

4x =

-3 + 7

4x =

x = 4

-3 - 7

4x =

x = - 2.5

The Quadratic Formula -b + √ b2 – 4ac

2ax =

2x = x2 - 3 ax2 + bx + c 2x = x2 - 3-2x -2x

0 = x2 – 2x - 3

0 = x2 – 2x - 3 ax2 + bx + c a = 1 b = -2 c = -3

-(-2) + √ (-2)2 – 4(1)(-3)

2(1)x =

-(-2) + √ (-2)2 – 4(1)(-3)

2(1)x =

-b + √ b2 – 4ac

2ax =

2 + √ 4 +12

2x = 2 + √ 16

2x =

2 + √ 16

2x = 2 + 4

2x =

2 + 4

2x =

x = 3

2 - 4

2x =

x = -1

Solving Quadratic Equations

• Graphing

• Factoring

• Using Square Roots

• Completing the Square

• Quadratic Formula

Solving Quadratic Equations

x + 4x - 2 = 0

x - 5x + 4 = 0

2

2