Algebra 2
Final Exam Review: Equations and Expressions
Solve each equation. Keep rational solutions in exact form, and round off irrational solutions
to the nearest hundredth.
1. 8(𝑥 + 3) − 2(𝑥 − 3) = 𝑥 + 6
8𝑥 + 24 − 2𝑥 + 6 = 𝑥 + 6
6𝑥 + 30 = 𝑥 + 6
5𝑥 = −24
24
𝑥=−5
2.
2
𝑥 = 40
5
5 2
5
∙ 𝑥 = 2 ∙ 40
2 5
𝑥 = 100
3.
5
6
3
𝑥+7=4
5
3
12 (6 𝑥 + 7 = 4)
10𝑥 + 84 = 9
10𝑥 = −75
𝑥 = −7.5
4. Solve by factoring.
𝑥 2 + 11𝑥 + 14 = 74
𝑥 2 + 11𝑥 − 60 = 0
(𝑥 + 15)(𝑥 − 4) = 0
𝑥 = −15, 𝑥 = 4
5. Solve by factoring.
8𝑥 2 = 20𝑥
8𝑥 2 − 20𝑥 = 0
4𝑥(2𝑥 − 5) = 0
5
𝑥 = 0, 𝑥 = 2
6. Solve by completing the square.
𝑥 2 − 10𝑥 + 85 = 11
𝑥 2 − 10𝑥 + ______ = −74 + _______
𝑥 2 − 10𝑥 + 25 = −74 + 25
(𝑥 − 5)2 = −49
𝑥 − 5 = ±7𝑖
𝑥 = 5 ± 7𝑖
7. Solve using the quadratic formula
3𝑥 2 + 5𝑥 + 8 = 7
3𝑥 2 + 5𝑥 + 1 = 0
𝑥=
−5±√52 −4(3)(1)
2(3)
−5±√13
𝑥= 6
𝑥 ≈ −0.23, 𝑥 ≈ −1.43
8. 𝑥 3 + 5𝑥 2 − 16𝑥 = 80
𝑥 3 + 5𝑥 2 − 16𝑥 − 80 = 0
𝑥 2 (𝑥 + 5) − 16(𝑥 + 5) = 0
(𝑥 + 5)(𝑥 2 − 16) = 0
(𝑥 + 5)(𝑥 − 4)(𝑥 + 4) = 0
𝑥 = −5, 𝑥 = 4, 𝑥 = −4
9. 𝑥 4 − 𝑥 2 = 90
𝑥 4 − 𝑥 2 − 90 = 0
(𝑥 2 + 9)(𝑥 2 − 10) = 0
𝑥 2 + 9 = 0, 𝑥 2 − 10 = 0
𝑥 2 = −9, 𝑥 2 = 10
𝑥 = ±3𝑖, 𝑥 = ±√10 ≈ ±3.16
10. 2𝑥 4 + 130𝑥 = 2𝑥
2𝑥 4 + 128𝑥 = 0
2𝑥(𝑥 3 + 64) = 0
2𝑥(𝑥 + 4)(𝑥 2 − 4𝑥 + 16) = 0
𝑥 = 0, 𝑥 = −4, 𝑥 = 2 ± 2√3𝑖
11. |2𝑥 + 3| = 17
2𝑥 + 3 = −17
2𝑥 = −20
𝑥 = −10
2𝑥 + 3 = 17
2𝑥 = 14
𝑥=7
12. |6 − 5𝑥| = 21
6 − 5𝑥 = −21
−5𝑥 = −27
𝑥 = 5.4
6 − 5𝑥 = 21
−5𝑥 = 15
𝑥 = −3
13. 200(1.13)𝑥 = 1300
1.13𝑥 = 6.5
𝑥 = log1.13 6.5
𝑥 ≈ 15.315
𝑥 2 − 4𝑥 + 16 = 0
𝑥 2 − 4𝑥 + 4 = −16 + 4
(𝑥 − 2)2 = −12
𝑥 − 2 = ±√−12
𝑥 − 2 = ±2√3𝑖
14. 17500(0.85)𝑥 = 7000
0.85𝑥 = 0.4
𝑥 = log 0.85 0.4
𝑥 ≈ 5.638
Solve each system of equations using the method of your choice.
15. 𝑦 = 7𝑥 + 10
6𝑥 + 2𝑦 = 96
6𝑥 + 2(7𝑥 + 10) = 96
6𝑥 + 14𝑥 + 20 = 96
20𝑥 = 76
𝑥 = 3.8
𝑦 = 7𝑥 + 10
𝑦 = 7(3.8) + 10
𝑦 = 36.6
(3.8, 36.6)
𝑥 + 4(1) = 17
𝑥 + 4 = 17
𝑥 = 13
(13, 1)
16. 5𝑥 − 3𝑦 = 62
𝑥 + 4𝑦 = 17
5𝑥 − 3𝑦 = 62
−5𝑥 − 20𝑦 = −85
−23𝑦 = −23
𝑦=1
17. 𝑥 + 𝑦 − 𝑧 = −9
𝑥−𝑦+𝑧 =1
−𝑥 + 𝑦 + 𝑧 = 11
𝑥
−4
1
1 −1 −1 −9
[𝑦] = [ 1 −1 1 ] ∙ [ 1 ] = [ 1 ]
𝑧
−1 1
1
11
6
(𝑥 + 4)3
18. Expand the power:
(𝑥 + 4)(𝑥 + 4)(𝑥 + 4)
(𝑥 + 4)(𝑥 2 + 8𝑥 + 16)
𝑥 3 + 8𝑥 2 + 16𝑥 + 4𝑥 2 + 32𝑥 + 64
𝑥 3 + 12𝑥 2 + 48𝑥 + 64
(𝑥 2 + 3𝑥 + 5)2
19. Expand the power:
(𝑥 2 + 3𝑥 + 5)(𝑥 2 + 3𝑥 + 5)
𝑥 4 + 3𝑥 3 + 5𝑥 2 + 3𝑥 3 + 9𝑥 2 + 15𝑥 + 5𝑥 2 + 15𝑥 + 25
𝑥 4 + 6𝑥 3 + 19𝑥 2 + 30𝑥 + 25
20. Simplify completely:
𝑥
4
𝑥2
𝑥
𝑥+2
4
4
− 𝑥 2 +2𝑥
𝑥 2 −4
− 𝑥(𝑥+2) = 𝑥(𝑥+2) − 𝑥(𝑥+2) = 𝑥(𝑥+2) =
𝑥+2
(𝑥+2)(𝑥−2)
𝑥(𝑥+2)
=
𝑥−2
𝑥
(−4,1,6)
21. Simplify completely:
6𝑥+4
𝑥−3
2
3+
𝑥
𝑥(𝑥−3)
6𝑥+4
𝑥−3
2
3+
𝑥
6𝑥 2 +4𝑥
2𝑥(3𝑥+2)
2𝑥(3𝑥+2)
2𝑥
∙ 𝑥(𝑥−3) = 3𝑥 2 −9𝑥+2𝑥−6 = 3𝑥 2 −7𝑥−6 = (3𝑥+2)(𝑥−3) = 𝑥−3
Factor completely.
22. 4𝑥 5 − 500𝑥 2
4𝑥 2 (𝑥 3 − 125)
4𝑥 2 (𝑥 − 5)(𝑥 2 + 5𝑥 + 25)
23. 𝑥 4 − 37𝑥 2 + 36
(𝑥 2 − 36)(𝑥 2 − 1)
(𝑥 + 6)(𝑥 − 6)(𝑥 + 1)(𝑥 − 1)
24. 15𝑥 3 − 114𝑥 2 − 360𝑥
3𝑥(5𝑥 2 − 38𝑥 − 120)
3𝑥(5𝑥 + 12)(𝑥 − 10)
25. 𝑥 5 − 100𝑥 3 + 1000𝑥 2 − 100000
𝑥 3 (𝑥 2 − 100) + 1000(𝑥 2 − 100)
(𝑥 2 − 100)(𝑥 3 + 1000)
(𝑥 + 10)(𝑥 − 10)(𝑥 + 10)(𝑥 2 − 10𝑥 + 100)