Warm–up #9
1. Solve by Factoring 2𝑥 3 − 15𝑥 2 = −28𝑥
2. Solve by Completing the Square
2
2𝑥 − 3𝑥 − 2 = 0
Solve by Factoring
1. 2𝑥 3 − 15𝑥 2 = −28𝑥
2𝑥 3 − 15𝑥 2 + 28𝑥 = 0
𝑥 2𝑥 2 − 15𝑥 + 28 = 0
𝑥 2𝑥 2 − 8𝑥 − 7𝑥 + 28 = 0
x(2x(x – 4) – 7(x – 4)) = 0
x(2x – 7)(x – 4) = 0
x=0
{0, 4,
2x – 7 = 0 x – 4 = 0
7
}
2
Make = 0!!!
Common factor first
2 #s that mult to
56
–7
–8
–15
& add to
set each factor = 0
Solve by Completing the
Square
2. 2𝑥 2 − 3𝑥 − 2 = 0
2
2
2
3
𝑥
2
𝑥 −
(𝑥 −
2
−1=0
9
3
𝑥 2 − 𝑥+ 16 = 1 +
2
2
−3
−3 1 2
÷2 =
∙
2
2 2
−3 2
9
=
=
4
16
9
16
3 2 25
) =
4
16
𝑥−
3
4
5
±
4
=
𝑥=
3
4
5
±
4
2, −
1
2
Homework Log
Wed
Learning Objective:
To solve quadratic forms
10/14
Lesson
2–4
Hw: #210 Pg. 128 #12, 24, 36,
37 – 51 odd, WS # 8 – 10
10/14/15 Lesson 2 – 4 Quadratic
Form Day 2
Advanced Math/Trig
Learning Objective

To solve quadratic equations
in quadratic form
Solve by Factoring
4
2
1. 4𝑥 = 13𝑥 − 9
4𝑥 4 − 13𝑥 2 + 9 = 0
Make = 0 first
It has quadratic form
2
4𝑢2 − 13𝑢 + 9 = 0
2
Let 𝑢 = 𝑥 2
4𝑢 − 13𝑢 + 9 = 0
4𝑢 − 4𝑢 − 9𝑢 + 9 = 0
4u(u – 1) – 9(u – 1) = 0
(4u – 9)(u – 1) = 0
2 #s that mult to
36
–4
–9
–13
& add to
Solve by Factoring
1. (4u – 9)(u – 1) = 0
2
Now replace u with 𝑥 2
2
(4𝑥 −9)(𝑥 − 1) = 0
(2x – 3)(2x + 3)(x + 1)(x – 1) = 0
2x – 3 = 0 2x + 3 = 0 x + 1 = 0 x – 1 = 0
3
± , ±1
2
4 Answers!!
Highest Power is 4!!
Solve by Factoring
6
3
2. 𝑥 + 7𝑥 − 8 = 0
Let 𝑢 = 𝑥 3
𝑢2 + 7𝑢 − 8 = 0
𝑢2 + 8𝑢 − 𝑢 − 8 = 0
u(u + 8) – 1(u + 8) = 0
(u – 1)(u + 8) = 0
2 #s that mult to
–8
8
–1
7
& add to
Solve by Factoring
2. (u – 1)(u + 8) = 0
Now replace u with 𝑥 3
(𝑥 3 −1)(𝑥 3 + 8) = 0
Diff & Sum of Cubes!
(𝑥 − 1)(𝑥 2 + 𝑥 + 1)(𝑥 + 2)(𝑥 2 − 2𝑥 + 4)
Solve by completing the square!
𝑥 2 + 𝑥+
1 2
2
=
1
4
1
4
= −1 +
1
4
𝑥 2 − 2𝑥+ 1 = −4 + 11
−2 2
2
= (−1)2 = 1
Solve by Factoring
2. (𝑥 − 1)(𝑥 2 + 𝑥 + 1) 𝑥 + 2 𝑥 2 − 2𝑥 + 4
𝑥+
1 2
2
1
2
x+ =
x=
1
−
2
=
3
−
4
𝑖 3
±
2
±
𝑖 3
2
𝑥−1
2
= −3
6 Answers!! x – 1 = ±𝑖 3
𝑥6
1 𝑖 3
− ±
, 1 ± 𝑖 3 , 1 , −2
2
2
x=1±𝑖 3
Solve by Factoring
3.
2
𝑥2
7
𝑥
Let 𝑢 =
+ +3=0
1
𝑥
2𝑢2 + 7𝑢 + 3 = 0
2
2𝑢 + 6𝑢 + 𝑢 + 3 = 0
2u(u + 3) + 1(u + 3) = 0
(2u + 1)(u + 3) = 0
2 #s that mult to
6
6
1
7
& add to
Solve by Factoring
3. (2u + 1)(u + 3) = 0 Now replace u with
2
2
𝑥
2
𝑥
1
𝑥
+1
1
𝑥
+3 =0
+1=0
1
𝑥
= −1
1
𝑥
2 = –x
+3=0
= −3
1 = –3x
1
𝑥
2 Answers!!
𝑥2
1
− , −2
3
Solve by Factoring
2
4. (𝑥 − 4) +2 𝑥 − 4 = 63
Let 𝑢 = 𝑥 − 4
(𝑥 − 4)2 +2 𝑥 − 4 − 63 = 0
𝑢2 + 2𝑢 − 63 = 0
2
𝑢 − 7𝑢 + 9𝑢 − 63 = 0
u(u – 7) + 9(u – 7) = 0
(u – 7)(u + 9) = 0
–63
–7 9
2 #s that mult to
2
& add to
Solve by Factoring
4. (u – 7)(u + 9) = 0
((x – 4) – 7)((x – 4) + 9) = 0
Now replace u
with 𝑥 − 4
(x – 11)(x + 5) = 0
x – 11 = 0
{– 5, 11}
x+5=0
2 Answers!!
Highest Power is 2!!
Solve by Factoring
4
5. 𝑥 = 16
4
𝑥 − 16 = 0
Let 𝑢 = 𝑥 2
±2𝑖, ±2
𝑢2 − 16 = 0
(u – 4) (u + 4) = 0
(𝑥 2 − 4)(𝑥 2 + 4) = 0
(x – 2)(x + 2)(𝑥 2 + 4) = 0
Diff of Squares
Replace u with 𝑥 2
𝑥2 + 4 = 0
2
𝑥 = −4
x = ±2𝑖
Solve by Factoring
2
6. 4(3𝑥 − 2) + 24 3𝑥 − 2 + 35 = 0
Let 𝑢 = 3𝑥 − 2
4𝑢2 + 24𝑢 + 35 = 0
4𝑢2 + 14𝑢 + 10𝑢 + 35 = 0
2u(2u + 7) + 5(2u + 7) = 0
(2u + 7)(2u + 5) = 0
140
14 10
2 #s that mult to
24
& add to
Solve by Factoring
6. (2u + 7)(2u + 5) = 0
(2(3x – 2) + 7)(2(3x – 2) + 5) = 0
(6x – 4 + 7)(6x – 4 + 5) = 0
(6x + 3)(6x + 1) = 0
6x + 3 = 0
1
1
− ,−
2
6
6x + 1 = 0
Replace u
with 3𝑥 − 2
Solve by Factoring
4
2
7. 60𝑥 = 147𝑥 − 27
60𝑥 4 − 147𝑥 2 + 27 = 0
Make = 0 first
Factor out GCF!
Let 𝑢 = 𝑥 2
2
3(20𝑢 − 49𝑢 + 9) = 0
2
3(20𝑢 − 45𝑢 − 4𝑢 + 9) = 0
3(5u(4u – 9) – 1(4u – 9) =
3(4u – 9)(5u – 1) = 0
180
0
–45 –4
2 #s that mult to
–49
& add to
Solve by Factoring
7. 3(4u – 9)(5u – 1) = 0
2
Now replace u with 𝑥 2
2
3(4𝑥 −9)(5𝑥 − 1) = 0
3(2x – 3)(2x + 3)(5𝑥 2 − 1) = 0
2
2x – 3 = 0 2x + 3 = 0 5𝑥 − 1 = 0
5𝑥 = 1
2
±
5
3
,±
5
2
𝑥 =
2
1
5
𝑥 =±
1
5
x=±
5
5
∙
5
5
Ticket Out the Door

Solve
Homework
#210 Pg. 128 #12, 24, 36, 37 – 51 odd,
WS # 8 – 10