CRASH COURSE IN QUADRATICSIn preparation for the Algebra CST
-b + b2 – 4ac
2ac√
(x+4)(x-3)=0
(x+1)(x+2)
X2 – 5x +4
F O I L
CompleteThe Square
Multiplying PolynomialsArea 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 PolynomialsArea 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 PolynomialsFOIL ( 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
Factoring Polynomials
3 412
72 5
10
76 1
6
77 2
14
9
3 5 6 418
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 + 36 13
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, 02 22 - 4(2) -4 2, -44 42 – 4(4) 0 4, 0
The Solution is the ________________
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 RootsSquare-Root Property
x2 = 16
√x2 = √16
4x2 – 25 = 0
x = +4
+25 +25√4x2 = √25
2x = 52 2x = + 2.5
4x2 = 25
x2 = 16(4)2= 16
(-4)2 = 16
Completing the SquareUsing Algebra Tiles x2 + 6x a= 1 b=6 c=0
b2( )2 ( )6
22
x2 + 6x = 0 b = 6+ 9 + 9
x2 + 6x + 9 = 9
(x+3)(x+3)=9(x+3)2 = 9
√(x+3)2 = √ 9x+3 = 3
x+3= 3+
x+3= -3x = 0x = -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 = √ 64x+7 = 8
x+7= 8+
x+7= -8x = 1x = -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 Squarex - 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 Formulax2 + 5x + 6 ax2 + bx + c a = 1
b = 5c = 62x2 + 3x – 5 = 0
ax2 + bx + ca = 2 b = 3 c = -5
-b + √ b2 – 4ac2a
x =
-b + √ b2 – 4ac2a
x =
-3 + √ 32 – 4(2)(-5)2(2)
x =
-3 + √ 9 – (-40)4
x = -3 + √ 494
x =
-3 + 74
x =
-3 + 74
x =
x = 4
-3 - 74
x =
x = - 2.5
The Quadratic Formula -b + √ b2 – 4ac2a
x =
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 – 4ac2a
x =
2 + √ 4 +122
x = 2 + √ 16
2x =
2 + √ 16
2x = 2 + 4
2x =
2 + 42
x =
x = 3
2 - 42
x =
x = -1
Solving Quadratic Equations
• Graphing
• Factoring
• Using Square Roots
• Completing the Square
• Quadratic Formula