Post on 23-Jul-2020
Alg. 2 Honors – Unit 3 Notes – Algebra behind Solving Quadratics
Factoring by Greatest Common Factor and Grouping
Objectives: SWBAT factor out a Greatest Common Factor from polynomials.
SWBAT identify polynomials that are prime.
SWBAT factor polynomials by grouping.
SWBAT use the zero product property to solve a quadratic.
GCF – Greatest Common Factor Prime – the GCF is only 1
A number/quantity that two terms
or more share
Find the greatest common factor.
1. – 6 and – 15 2. 16𝑥, 24𝑥 and 36𝑥 3. 3𝑥2 and 12𝑥
–3 4x 3x
Factor out the greatest common factor.
1 – Find the GCF, 2 – Divide it out, and 3 – Write as GCF (Remainder)
4. 6𝑥 − 14 5. −4𝑥2𝑦 − 6𝑥𝑦2 6. 7𝑥2 − 28𝑥 + 14
6 14
2
2 3 7
x
x
2 24 6
2
2 2 3
x y xy
xy
xy x y
2
2
7 28 14
7
4 2
x x
y x x
7. 8𝑥2 + 9𝑧2 8. 18𝑤3 − 15𝑤4 + 5𝑤5 9. (2𝑥2)(4𝑥2 − 12𝑥)
Prime
3 4 5
3
3 2
18 15 5
18 15 5
w w w
w
w w w w
2
2
3
4 12
4
4 3
2 4 3
8 3
x x
x
x x
x x x
x x
Factor by Grouping
Factor our a ( ) they two terms have in common
OR
Put ( ) around the 1st two terms and last two terms
Factor out a GCF of each parenthesis
They should have a ( ) in common
10. 𝑎(𝑏 + 4) + 𝑐(𝑏 + 4) 11. 𝑎(𝑥 − 3) + 𝑥2(𝑥 − 3) 12. 𝑥2 − 3𝑥 + 4𝑥 − 12
4 4
4
4
a b c b
b
b a c
2
2
3 3
3
3
a x x x
x
x a x
2
2
3 4 12
3 4 12
4
3 4 3
3 4 3
3
4 3
x x x
x x x
x
x x x
x x x
x
x x
13. 3𝑦2 − 8𝑦 + 12𝑦 − 32 14. 𝑎𝑏 + 𝑏𝑐 + 𝑎 + 𝑐 15. 3𝑥2 + 3𝑥𝑦 − 2𝑥𝑦 − 2𝑦2
2
2
3 8 12 32
3 8 12 32
4
3 8 4 3 8
3 8 4 3 8
3 8
3 8 4
y y y
y y y
y
y y y
y y y
y
y y
1
1
1
1
ab bc a c
ab bc a c
b
b a c a c
b a c a c
a c
a c b
2 2
2 2
3 3 2 2
3 3 2 2
3 2
3 2
3 2
3 2
x xy xy y
x xy xy y
x y
x x y y x y
x x y y x y
x y
x y x y
ZERO PRODUCT PROPERTY
0
0 and/or 0
a b
then
a b
Finding the Z.A.R.S.
Zeros~ Answers~ Roots~ Solutions~
When the directions ask you to find the Zeroes, Answers, Roots, or Solutions – you first
factor then set each parenthesis equal to zero and solve each parenthesis.
Solve the following.
16. (𝑥 + 3)(𝑥 − 5) = 0 17. 𝑦(4𝑦 − 9) = 0 18. 𝑎2 − 12𝑎 = 0
3 0 5 0
3 5
x x
x x
4 9 0
0 4 9 0
93
4
y y
y y
y y
2 12
12 0
12 0
0 12 0
0 12
Factor
a a
a
a a
Solve
a a
a a
x x
19. 5𝑧2 = 30𝑧 20. 7𝑚2 − 15𝑚 = −25𝑚2 21. 24 12 3 9 0m m m
2
2
5 30 0
5 30
5
5 6 0
5 6 0
5 0 6 0
0 6
z z
Factor
z z
z
z z
Solve
z z
z z
z z
2
2
32 12 0
32 12
4
4 8 3 0
4 8 3 0
4 0 8 3 0
80
3
m m
Factor
m m
m
m m
Solve
m m
m m
m m
Not Factorable
22. 2 5 5 25 0y y y 23. 28 6 12 9 0x x x 24. 27 21 8 24t t t
2
2
5 5 25
5 5 25
5
5 5 5
5 5 5
5
5 5
5 5 0
5 0 5 0
5 5
Factor
y y y
y y y
y
y y y
y y y
y
y y
Solve
y y
y y
y y
2
2
8 6 12 9
8 6 12 9
2 3
2 4 3 3 4 3
2 4 3 3 4 3
4 3
4 3 2 3
4 3 2 3 0
4 3 0 2 3 0
3 3
4 2
Factor
x x x
x x x
x
x x x
x x x
x
x x
Solve
x x
x x
x x
2
2
2
7 21 8 24 0
7 21 8 24
7 21 8 24
7 8
7 3 8 3
7 3 8 3
3
3 7 8
3 7 8 0
3 0 7 8 0
83
7
t t t
Factor
t t t
t t t
t
t t t
t t t
t
t t
Solve
t t
t t
t x
Factoring Trinomials when a > 1
Objectives: SWBAT factor “hard” trinomials, trinomials with a leading coefficient greater than 1.
SWBAT solve a quadratic equation with a leading coefficient greater than 1.
Factoring Trinomial when a > 1
Put the equation in Standard form Find the two numbers that multiply to Factor by grouping / box
the Special number and add to the B
Factor by Grouping
1. 22 9 7x x
Fill in the “x” Rewrite the polynomial Factor first
two terms
Factor last
two terms
22 72 7xx x
2
2
2 2
2 2
2
2 1
x x
x x
x
x x
7 7
7 7
7
7 1
x
x
x
Answer 2 7 1x x
2. 23 2x x
Fill in the “x” Rewrite the polynomial Factor first two terms
Factor last two terms
23 23 2xx x
2
2
3 3
3 3
3
3 1
x x
x x
x
x x
2 2
2 2
2
2 1
x
x
x
Answer 3 2 1x x
2ax bx c /
Grouping
Box Method
3. 23 7 20x x
Fill in the “x” Rewrite the polynomial Factor first
two terms
Factor last
two terms
2 5 20123 xx x
2
2
3 12
3 12
3
3 4
x x
x x
x
x x
5 20
5 20
4
4 4
x
x
x
Answer 3 4 4x x
Factor by X Box Method
6. 22 21y x x
Fill in the “x”
Fill in the box Factor out
Answer 2 7 3x x
7. 2( ) 4 25 6f x x x
Fill in the “x”
Fill in the box Factor out
Answer 4 1 6x x
A
Term
C
Term
1
Factor
2
Factor
A
Term
C
Term
1
Factor
2
Factor
22x
21
6x
7x
3x
2x
7
24x
6
24x
1x
6x
4x
1
ZERO PRODUCT PROPERTY
0
0 and/or 0
a b
then
a b
Finding the Z.A.R.S.
Zeros~ Answers~ Roots~ Solutions~
When the directions ask you to find the Zeroes, Answers, Roots, or Solutions – you first
factor then set each parenthesis equal to zero and solve each parenthesis.
Find the roots of ax2 + bx + c = 0 (By factoring and using the zero product rule).
9. 26 7 2x x 10. 2 8 12w w 11. 2 23 18 2 3 4x x x x
Put all equation in standard form first / Factor any way you would like
26 7 2 0x x 2 8 12 0w w 27 15 2x x
2
2
6 2
6 2
2 3 2 1 3 2
2 1 3 2
2 1 0 3 2 0
1 2
2 3
4 3
4 3
x
x
x x x
x x
x x
x
x x
x x
x
2
2
12
12
2 6 2
2 6
2 6
2 0 6 0
2
6
6
2
w
w
w w w
w w
w w
x x
w w
w w
27x
2
14x
1x
2x
7x
1
7 1 2 0
7 1 0 2 0
1 2
7
x x
x x
x x
Factoring “Easy” Trinomials (a = 1) and Factoring by Substitution
Objectives: SWBAT factor “easy” trinomials, trinomials with a leading coefficient of 1.
SWBAT solve a quadratic equation with a leading coefficient of 1.
SWBAT factor quadratics using substitution.
The “Short cut” when a=1
When a=1, you can go from the x straight to the ( ) when factoring
Factor trinomials in the form x2 + bx + c (using the x-factor).
1. x2 + 7x – 8 2. n2 + 7n +12 3. x2 – 4x – 12
1 8x x 3 4n n 2 6x x
Find the roots of ax2 + bx + c = 0 (By factoring and using the zero product rule).
4. x2 – 2x – 15 = 0 5. x2 + 2 = 3x 6. Find the zeros of:
𝑓(𝑥) = 𝑥2 + 3𝑥 − 40
2 3 2 0x x Zeroes just means your answer are
points
5 3 0
5 0 3 0
5 3
x x
x x
x x
2 1 0
2 0 1 0
2 1
x x
x x
x x
8 5 0
8 0 5 0
8 5
8,0 & 5,0
x x
x x
x x
7. 2x2 + 8x – 24 = 0 8. 4k2 + 14k = 30 9. 0 = 3x2 + 33x + 36
2
2
2
2 8 24 0
2 2
4 12 0
GCF
x x
x x
2
2
2
4 14 30 0
2
2 8 24 0
2 2
2 7 15 0
k k
GCF
k k
k k
2
2
3
3 33 36 0
3 3
11 12 0
GCF
x x
x x
Can’t go any further at this
Point (need Quadratic Formula)
U substitution
You can substitute a “u” for a ( ), and then factor the new statement using the u. Then at
the end of the factoring, back substitute the ( ) for the u.
Factor quadratics using substitution.
10. (x + 1)2 + 4(x + 1) + 4 11. (x – 2)2 + 4(x – 2) – 21
2
1
4 4
u x
u u
2
2
4 21
u x
u u
2 2
1
1 2 1 2
3 3
u u
u x
x x
x x
7 3
2
2 7 2 3
5 5
u u
u x
x x
x x
2
2
2 10 3 15
2 10 3 15
2 5 3 5
5 2 3
5 0 2 3 0
35
2
k k k
k k k
k k k
k k
k k
k k
Special Factoring Patterns
Objectives: SWBAT identify different types of factoring problems.
SWBAT identify and use special factoring patterns.
SWBAT factor the “un-factorable”.
Perfect Squares Addition Rule Subtraction Rule
2 2a b a b a b 2 22a b a b a ab b 2 22a b a b a ab b
The key important concept is the perfect square rule and being about to see that if a
binomial (two terms) can be factored.
Factor with special patterns.
1. x2 – 25 2. m2 – 22m + 121 3. d2 – 64 4. x2 + 12x + 36
Perfect Square Subtraction Rule Perfect Square Addition Rule
2
11 11
11
m m
or
m
2
6 6
6
x x
or
x
2
25
5 5
5 5
two numbers that make x
x and x
two numbers that make
and
x x
2
64
8 8
8 8
two numbers that make d
d and d
two numbers that make
and
d d
5. 9x2 – 25 6. 4x2 + 36 7. d2 – 12
2 9
3 3
25
5 5
3 5 3 5
two numbers that make x
x and x
two numbers that make
and
x x
2
9
3 3
3 3
two numbers that make x
x and x
two numbers that make
i and i
x i x i
2
2
4
4 36
4
4 9
GCF
x
x
2
12
12 12
12 2 3
2 3 2 3
two numbers that make d
d and d
two numbers that make
and
d d
Find the roots of the following equations.
8. x2 + 18 = 9x 9. 2x2 – 5x + 3 = 0 10. 4x2 + 36 = 0 11. x2 – 121 = 0
2 9 18 0x x
6 3 0
6 0 3 0
6 3
x x
x x
x x
12. 4x3 – 4x2 = 224x 13. 2x2 = 50 14. 3y3 – 12y2 – 2y2 + 8 = 0
2
2
2 2 3 3
2 2 3 3
2 1 3 1
2 3 1
2 3 0 1 0
3 1
2
x x x
x x x
x x x
x x
x x
x x
2
9
3 3
3 3 0
3 0 3 0
3 3
two numbers that make x
x and x
two numbers that make
i and i
x i x i
x i x i
x i x i
2
2
4
4 36 0
4 4
9 0
GCF
x
x
2
121
11 11
11 11 0
11 0 11 0
11 11
two numbers that make x
x and x
two numbers that make
and
x x
x x
x x
3 2
2
4 4 224 0
4 56
x x x
x x x
factor just inside
4 8 7
4 0 8 0 7 0
0 8 7
x x x
x x x
x x x
22 50 0
2 25
'
x
x x
can t factor just inside
go straight to solving
2 0 25 0
0 25
x x
x x
3 2 2
3 2 2
2
2
2
4
3 12 2 8
3 12 2 8
3 2
3 4 2 4
3 2 4
'
T erms Factor by Grouping
y y y
y y y
y
y y y
y y
can t factor just inside
go straight to solving
2
2
2
2
3 2 4
3 2 0 4 0
3 2
2
3
2 4
3
6 4
3
y y
y y
y
y
y y
y y
Solve Quadratic Equations by Finding Square Roots
Objective: SWBAT solve quadratic equations by using square roots.
Square root - Principal square root -
Radical - Radicand -
Simplify the radical expression.
1. 2. 3.
10 15
150
5 6
4. 5 6 2 18 5. 17
16
/ \
4 6
/ \
2 2
24
2 6
/ \
49 2
/ \
7 7
98
7 2
10 108
10 6 3
60 3
17
16
17
4
Rationalizing the denominator - Conjugates -
Can’t have a square root in the denominator Multiple terms in the denominator
Multiply the top and bottom to make a perfect square Multiply the top and bottom by the
on the bottom a perfect square of the denominator
Simplify the radical expression.
6. 6
5 7.
8
12 8.
2
4 5
Solving Quadratics by Square Roots
Isolate the variable or parenthesis squared
Take a square root of both sides (don’t forget the plus or minus if it is an even root) Solve for the variable
Use square roots to solve the following quadratic equations.
10. 3x2 = 75 11. z2 – 7 = 25
Reduce First
2 3
3 3
6
9
6
3
6 5
5 5
30
25
30
5
2 4 5
4 5 4 5
4 2 10
16 4 5 4 5 25
4 2 10
16 5
4 2 10
11
2
2
2
3 75
25
25
5
x
x
x
x
2
2
2
7 25
32
32
4 2
z
z
z
z
12. 2x
6 1025
13. 24(x 1) 8
2
2
2
2
6 1025
425
100
100
10
x
x
x
x
x i
2
2
2
4( 1) 8
( 1) 2
( 1) 2
1 2
1 2
x
x
x
x
x
Complete the Square
Objective: SWBAT complete the square.
SWBAT factor by completing the square.
SWBAT solve quadratics by completing the square.
SWBAT derive the quadratic formula.
Perfect square trinomials - 2
x a x a x a
𝒙𝟐 – 𝟏𝟎𝒙 + 𝟏𝟐 = 𝟎
Find the c value that would make the following a perfect trinomial.
1. 2x 10x c 2. 2x 8x c 3. 2x 13x c
2
2
2
2
2
10___ ___
2
+5
5
10 ___ 25 ___
10 25
5 2
xx c
x x
x x
x c
2
2
2
2
2
8___ ___
2
4
4
8 ___16 ___
8 16
4 16
xx c
x x
x x
x c
2
2
2
2
2
13___ ___
2
13
2
13
2
1698 ___ ___
4
16913
4
13 169
2 4
xx c
x x
x
x c
2
2
2
2
2
2
2
2
10 12
10 ______ 12 _______
10______ 12 _______
2
5
5
10 ___ 25 ___ 12 ___ 25 ____
10 25 13
5 13
5 13
5 13
5 13
x x
x x
xx
x x
x x
x
x
x
x
Solve the following by completing the square.
4. 2x 12x 18 0 5. 2x 10x 21 0
2
2
2
2
2
2
2
2
12 18
12 ______ 18 _______
12______ 18 _______
2
6
6
12 ___ 36 ___ 18 ___ 36 ____
12 36 18
6 18
6 18
6 3 2
6 3 2
x x
x
xx
x x
x x
x
x
x
x
2
2
2
2
2
2
2
2
10 21
10 ______ 21_______
10______ 21_______
2
5
5
10 ___ 36 ___ 21___ 25 ____
10 36 4
5 4
5 4
5 2
5 2
3
7
x x
x
xx
x x
x x
x
x
x
x
x
x
6. 2x 8x 1 7. 23x 42x 24
8. 2x 5x 12 0
2
2
2
2
2
2
2
5 12
5______ 12 ______
2
5
2
5
2
25 255 ___ ___ 12 __ ____
4 4
255
4
5 23
2 4
5 23
2 4
5 23
2 2
5 23
2 2
x x
xx
x x
x x
x
x
ix
ix
2
2
2
2
2
2
2
2
8 1
8 ______ 1_______
8______ 1_______
2
4
4
8 ___16 ___ 1___16 ____
8 16 15
4 15
4 15
4 15
4 15
x x
x
xx
x x
x x
x
x
x
x
2
2
2
2
2
2
2
2
2
3 42 24
3
14 8
14 ______ 8 _______
14______ 8 _______
2
7
7
14 ___ 49 ___ 8 ___ 49 ____
14 16 41
7 41
7 41
7 41
7 41
x x
x x
x x
xx
x x
x x
x
x
x
x
Use the Quadratic Formula and the Discriminant
Objective: SWBAT find and use the discriminant.
SWBAT use the quadratic formula.
Quadratic Formula - Discriminant -
2 4
2
b b acx
a
2 4b ac
Value of the
discriminant Number and type of solutions
Graph
Zero
1 real solution
Positive discriminant
2 real solutions
Negative discriminant
2 imaginary solutions
Find the discriminant and describe the solutions.
1. 23x 11x 4 0 2. 22a 13a 21
2
2
3
11
4
4
11 4 3 4
121 48
73
2
a
b
c
b ac
real solutions
2
2
2
13
21
4
13 4 3 21
169 252
83
2 imaginary
a
b
c
b ac
solutions
Use the quadratic formula to solve the following equations. Give exact answers, no decimals.
1. 2x 12x 32 0 2. 2x 4x 1 0
2
2
1
12
32
4
2
12 12 4 1 32
2 1
12 16
2
12 4
2
6 2
4 8
a
b
c
b b acx
a
x
x
x
x
x x
2
2
1
4
1
4
2
4 4 4 1 1
2 1
4 12
2
4 2 3
2
2 3
a
b
c
b b acx
a
x
x
x
x
3. 23x 11x 4 0 4. 22a 13a 7
2
2
3
11
4
4
2
11 11 4 3 4
2 3
11 73
6
11 73
6
a
b
c
b b acx
a
x
x
x
2
2
2
13
7
4
2
13 13 4 2 7
2 2
13 225
4
13 15
4
17
2
a
b
c
b b acx
a
x
x
x
x x
5. 2x x 1 0 6. 2x 12x 1 4x 3
2
2
1
1
1
4
2
1 1 4 1 1
2 1
1 3
2
1 3
2
a
b
c
b b acx
a
x
x
ix
2
2
2
16 4 0
1
16
4
4
2
16 16 4 1 4
2 1
16 240
2
16 4 15
2
8 2 15
x x
a
b
c
b b acx
a
x
x
x
x