In preparation for the Algebra CST b b 2 4ac 2ac x 4x30 x 1x 2 X 2 5x 4 F O I L Complete The Square Multiplying Polynomials Area Model of Multiplication ID: 695171
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Slide1
CRASH COURSE IN QUADRATICS
In preparation for the Algebra CST
-b
+
b2 – 4ac2ac
√
(x+4)(x-3)=0
(x+1)(x+2)
X
2
– 5x +4
F O I L
Complete
The SquareSlide2
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 53Slide3
Multiplying Polynomials
Area Model of Multiplication
(x)(x)
x
2
(x)(2)
2x
(3)(x)
3x
(3)(2)
6
x + 2
x + 3
X
2
+ + 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)Slide4
Multiplying Polynomials
FOIL
( x + 2 ) ( x + 3)
F
irst
(x)(x) = x
2
O
uter
(x)(3) = 3x
I
nner
(2)(x) = 2x
L
ast
(2)(3) = 6
Combine like terms…
=
x
2
+ 5x + 6
Slide5
Multiplying Polynomials
x
2 + 5x + 6
a
x2 + bx + c
a = 1b = 5c = 6Slide6
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?”Slide7
Factoring Polynomials
X
2 + 7x + 12
7
12
(x + )(x+ )
X
2
+ 13x + 36
13
36
(x + )(x+ )
(x + )(x+ )
-6
-40
X
2
- 6x - 40Slide8
Perfect Square Trinomial
X
2 + 12 + 36
X * X
6 * 6
(x + 6)(x + 6)
(x + 6)
2
X
2
- 14 + 49
X * X
7 * 7
(x - 7)(x - 7)
(x - 7)
2
-Slide9
Solving Quadratic Equations
Graphing
Factoring
Using Square Roots
Completing the SquareQuadratic FormulaSlide10
Graphing Quadratic Equations
x
2 – 4x = 0
x
y=x2 - 4x
y x, y
0
0
2
– 4(0)
0
0, 0
2
2
2
- 4(2)
-4
2, -4
4
4
2
– 4(4)
0
4, 0
The Solution is the
________________Slide11
Find the solution for each graph:Slide12
Factoring Quadratic Equations
Using the
Zero Product Property
(x-3)(x+7)=0
(x-3)=0(x+7)=0
x = 3
x = -7Slide13
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)=0Can you solve in your head?(x-2)(x+1)=0
x2 + 12x + 36x =
x
2 - 21x = 72x =
-72
-21
If
x
2
is added to
x
, the sum is 42. What are the values of
x
? Slide14
Using Square Roots
Square-Root Property
x
2
= 16√x2 =
√16
4x2 – 25 = 0
x =
+
4
+25
+25
√
4x2 = √
25
2x = 5
2 2
x =
+
2.5
4x
2
= 25
x
2
= 16
(4)
2
= 16
(-4)
2
= 16Slide15
Completing the Square
Using Algebra Tiles
x
2
+ 6xa= 1 b=6 c=0
b2
( )
2
( )
6
2
2
x
2 + 6x = 0
b = 6
+ 9
+ 9
x
2
+ 6x + 9 = 9
(x+3)(x+3)=9
(x+3)
2
= 9
√
(x+3)
2
=
√
9
x+3
= 3
x+3= 3
+
x+3= -3
x = 0
x = -6
( )
6
2
2
= 9Slide16
Completing the Square
x
2 + 14x = 15
b = 14
14 2
( )
2
= 7
2
=49
Add to both sides of the equation
+ 49
+ 49
x
2
+ 14x + 49 = 64
Factor the Perfect Square
(x+7)(x+7)=64
(x+7)
2
= 64
√
(x+7)
2
=
√
64
x+7
= 8
x+7= 8
+
x+7= -8
x = 1
x = -15Slide17
Completing the Square
x
2 - 10x
= -1 3
b = 10 3
Add to both sides of the equation
Factor the Perfect Square
3x
2
– 10x = -3
3
3
3
-
10
1
3 2
( )
2
*
=
100
36
Reduce
25
9
x
2
-
10x
= -1
3
+
25
9
+
25
9
-
9
+
25
9 9
=
16
9
x
2
-
10x
+
25
=
16
3 9 9
x –
5
3
( )
4
3
=
+
x –
5
3
√
( )
2
16
9
=
√
x –
5
=
4
3 3
x –
5
= -
4
3 3
x =
9
3
x =
1
3
x –
5
3
( )
16
9
=
2Slide18
Completing the Square
x - 8x = 12
x - 8x = 5
What should be added to both sides of this equation?
x + 4x = 6x - 4x = 8ax – bx = c
2
2
2
2
2Slide19
The Quadratic Formula
x
2 + 5x + 6
a
x2 + bx + c
a = 1b = 5c = 6
2x
2 + 3x – 5 = 0
ax
2 + bx +
c
a = 2 b = 3 c = -5
-b
+
√
b
2
– 4ac
2a
x =
-b
+
√
b
2
– 4ac
2a
x =
-3
+
√
3
2
– 4(2)(-5)
2(2)
x =
-3
+
√
9 – (-40)
4
x =
-3
+
√
49
4
x =
-3
+
7
4
x =
-3 + 7
4
x =
x = 4
-3 - 7
4
x =
x = - 2.5Slide20
The Quadratic Formula
-b
+
√ b2 – 4ac2a
x =
2x = x
2
- 3
ax2 +
bx + c
2x = x2 - 3
-2x
-2x
0 = x
2
– 2x - 3
0 = x
2
– 2x - 3
a
x
2
+
b
x
+ 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
+
√
b
2
– 4ac
2a
x =
2
+
√ 4
+12
2
x =
2
+
√ 16
2
x =
2
+
√ 16
2
x =
2
+
4
2
x =
2 +
4
2
x =
x = 3
2 -
4
2
x =
x = -1Slide21
Solving Quadratic Equations
Graphing
Factoring
Using Square Roots
Completing the SquareQuadratic FormulaSlide22
Solving Quadratic Equations
x + 4x - 2 = 0
x - 5x + 4 = 0
2
2