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
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
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
Perfect Square Trinomials
Create perfect square trinomials.
x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___
100
4
25/4
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
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
Solving Quadratic Equations by Completing the Square
Step 4: Take the square root of each side
2( 4) 36x
( 4) 6x
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
Solving Quadratic Equations by
Completing the Square
Section 8.1
Completing the Square
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
Square Root PropertySquare Root Property
If x and b are complex numbers and if x 2 = b, then
bx OR
bx
Solve each equation. Write radicals in simplified form.
492 k
49k
7k
Square Root Property
Solve each equation. Write radicals in simplified form.
112 b
11b Square Root Property
Radical will not simplify.
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
Solve each equation by factoring. 3x2 =5x
Homework Quiz
Solve each equation by factoring. 3x2 =5x
x= {0, 5/3}
Homework Quiz
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
Solve each equation. Write radicals in simplified form.
122 c
12c
32c Solution Set
Square Root Property
Solve each equation. Write radicals in simplified form.
36)2( 2 m
36)2( m62 m
62 m22 8m62 m22
4m}4,8{m
Solve each equation. Write radicals in simplified form.
3)4( 2 p3)4( p
34 p
Perfect Square Trinomials
Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36
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
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
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
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
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
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
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
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
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
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
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
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
