Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3}...
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Transcript of Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3}...
![Page 1: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/1.jpg)
FactoringFactoring x2 = 9
x2 - 9 = 0(x + 3)(x - 3) = 0
x + 3 = 0 or x - 3 = 0x = -3 or x = 3
x = {-3, 3}
Zero-factorproperty
![Page 2: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/2.jpg)
Another Way to Solve QuadraticsSquare Root Property
When you introduce the radicalyou must use + and - signs.
Recall that we know thesolution set is
x = {-3, 3}
92 x
92 x3x
92 x
92 x
3x
![Page 3: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/3.jpg)
Solving Quadratic Equations by Completing the Square
Solve the following equation by completing the square:
Step 1: Move quadratic term, and linear term to left side of the equation
2 8 20 0x x
2 8 20x x
![Page 4: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/4.jpg)
Perfect Square Trinomials
Create perfect square trinomials.
x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___
100
4
25/4
![Page 5: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/5.jpg)
Creating a Perfect Square Trinomial
In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____
Find the constant term by squaring half the coefficient of the linear term.
(14/2)2
X2 + 14x + 49
![Page 6: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/6.jpg)
Solving Quadratic Equations by Completing the Square
Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.
2 8 =20 + x x 21
( ) 4 then square it, 4 162
8
2 8 2016 16x x
![Page 7: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/7.jpg)
Solving Quadratic Equations by Completing the Square
Step 4: Take the square root of each side
2( 4) 36x
( 4) 6x
![Page 8: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/8.jpg)
Solving Quadratic Equations by Completing the Square
Step 5: Set up the two possibilities and solve
4 6
4 6 an
d 4 6
10 and 2 x=
x
x x
x
![Page 9: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/9.jpg)
Solving Quadratic Equations by
Completing the Square
![Page 10: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/10.jpg)
Section 8.1
Completing the Square
![Page 11: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/11.jpg)
FactoringFactoring Before today the only way we had for
solving quadratics was to factor.
x2 - 2x - 15 = 0(x + 3)(x - 5) = 0
x + 3 = 0 or x - 5 = 0x = -3 or x = 5
x = {-3, 5}
Zero-factorproperty
![Page 12: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/12.jpg)
Square Root PropertySquare Root Property
If x and b are complex numbers and if x 2 = b, then
bx OR
bx
![Page 13: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/13.jpg)
Solve each equation. Write radicals in simplified form.
492 k
49k
7k
Square Root Property
![Page 14: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/14.jpg)
Solve each equation. Write radicals in simplified form.
112 b
11b Square Root Property
Radical will not simplify.
![Page 15: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/15.jpg)
AAT-A Date: 2/5/14 SWBAT complete the square to solve factoring problems Do Now: HW Requests: pg 303 #42-49; Pg 310 #15-37 odds
In Class: Start Completing the Square WSHW: Complete WS KutaSoftware 1-24 oddsBegin Section 6.5Announcements: •Tutoring: Tues. and Thurs. 3-4•Bring Graphing Calculator toClass for Thursday•Quiz Friday w/HW Quiz before•Complete presentations
Life Is Just A MinuteLife is just a minute—only sixty seconds in it.Forced upon you—can't refuse it.Didn't seek it—didn't choose it.But it's up to you to use it.You must suffer if you lose it.Give an account if you abuse it.Just a tiny, little minute,But eternity is in it!
By Dr. Benjamin Elijah Mays, Past President of Morehouse College
![Page 16: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/16.jpg)
Solve each equation by factoring. 3x2 =5x
Homework Quiz
![Page 17: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/17.jpg)
Solve each equation by factoring. 3x2 =5x
x= {0, 5/3}
Homework Quiz
![Page 18: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/18.jpg)
Solving Quadratic Equations by Completing the Square
2
2
2
2
2
1. 2 63 0
2. 8 84 0
3. 5 24 0
4. 7 13 0
5. 3 5 6 0
x x
x x
x x
x x
x x
Try the following examples. Do your work on your paper and then check your answers.
1. 9,7
2.(6, 14)
3. 3,8
7 34.
2
5 475.
6
i
i
![Page 19: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/19.jpg)
Solve each equation. Write radicals in simplified form.
122 c
12c
32c Solution Set
Square Root Property
![Page 20: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/20.jpg)
Solve each equation. Write radicals in simplified form.
36)2( 2 m
36)2( m62 m
62 m22 8m62 m22
4m}4,8{m
![Page 21: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/21.jpg)
Solve each equation. Write radicals in simplified form.
3)4( 2 p3)4( p
34 p
![Page 22: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/22.jpg)
Perfect Square Trinomials
Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36
![Page 23: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/23.jpg)
Completing the Square
1. Divide by the coefficient of the squared term. Make the coefficient of the squared term =1.2. Move all variables to one side and constants to the other.3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.4. Factor the left hand side and simplify the right.5. Root and solve.
2 2 24 0x x
1 1 1 12 2 24 0x x
2 2 24x x
2
12 1
2
1 1
1
21 25x
1
21 25x
![Page 24: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/24.jpg)
Completing the Square
1.Divide by the coefficient of the squared term. Make the coefficient of the squared term =1.2. Move all variables to one side and constants to the other.3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.4. Factor the left hand side and simplify the right.5. Root and solve.
21 25x
1 5x
1 5x { 4, 6}x
![Page 25: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/25.jpg)
Completing the Square
1. Make the coefficient of the squared term =1.
2. Move all variables to one side and constants to the other.
3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.
4. Factor the left hand side and simplify the right.
5. Root and solve.
0253 2 xx3333
03
2
3
52 xx
3
2
3
52 xx1 5 5
2 3 6
2 25 5
6 6
36
49
6
52
x
2 25 5
6 6
![Page 26: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/26.jpg)
Completing the Square
1.Divide by the coefficient of the squared term. Make the coefficient of the squared term =1.2. Move all variables to one side and constants to the other.3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.4. Factor the left hand side and simplify the right.5. Root and solve.
25 49
6 36x
36
49
6
5x
6
7
6
5x
26
12
6
7
6
5
x
3
1
6
2
6
7
6
5x
3
1,2x
![Page 27: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/27.jpg)
1. Make the coefficient of the squared term =1.
2. Move all variables to one side and constants to the other.
3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.
4. Factor the left hand side and simplify the right.
5. Root and solve.
2 5 3 0x x 2 5 3x x
1 5 5
2 1 2
252
25 37
2 4x
252
5 37
2 2x
5 37
2x
![Page 28: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/28.jpg)
Solving Quadratic Equations Solving Quadratic Equations by Completing the Squareby Completing the Square
x2 - 2x - 15 = 0(x + 3)(x - 5) = 0x + 3 = 0 or x - 5 = 0x = -3 or x = 5x = {-3, 5}
01522 xx 1522 xx
Now take 1/2 of the coefficient of x.
Square it.Add the result to both sides. 11
Factor the left.Simplify the right.
16)1( 2 x161 x41x}5,3{x
Square Root Property
![Page 29: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/29.jpg)
Solving Quadratic Equations by Completing the Square
Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.
2 8 2016 16x x
2
( 4)( 4) 36
( 4) 36
x x
x
![Page 30: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/30.jpg)
Deriving The Quadratic Formula
2 0b c
x xa a
Divide both sides by a
2 22
2 2
b b c bx x
a a a a
2 2
2 2
4
2 4 4
b b acx
a a a
2
2
4
2 4
b b acx
a a
2 4
2
b b acx
a
Complete the square by adding (b/2a)2 to both sides
Factor (left) and find LCD (right)
Combine fractions and take the square root of both sides
Subtract b/2a and simplify
2If 0 (and 0 then:),ax bx c a
![Page 31: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/31.jpg)
Completing the Square-Example #2
Solve the following equation by completing the square:
Step 1: Move quadratic term, and linear term to left side of the equation, the constant to the right side of the equation.
22 7 12 0x x
22 7 12x x
![Page 32: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/32.jpg)
Solving Quadratic Equations by Completing the Square
Step 2: Find the term that completes the square on the left side of the equation. Add that term to both sides.
The quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first.
2
2
2
2 7
2
2 2 2
7 12
7
2
=-12 +
6
x x
x x
xx
21 7 7 49
( ) then square it, 2 62 4 4 1
7
2 49 49
16 1
76
2 6x x
![Page 33: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/33.jpg)
Solving Quadratic Equations by Completing the Square
Step 3: Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.
2
2
2
76
2
7 96 49
4 16 16
7 47
4
49 49
16 1
16
6x x
x
x
![Page 34: Factoring x 2 = 9 x 2 - 9 = 0 (x + 3)(x - 3) = 0 x + 3 = 0 or x - 3 = 0 x = -3 or x = 3 x = {-3, 3} Zero-factor property.](https://reader030.fdocuments.net/reader030/viewer/2022033007/56649de75503460f94ae1589/html5/thumbnails/34.jpg)
Solving Quadratic Equations by Completing the Square
Step 4: Take the square root of each side
27 47( )
4 16x
7 47( )
4 4
7 47
4 4
7 47
4
x
ix
ix