GP1$PAP$S2$HW4$Solutions2 Skill22b2Day22 ...
Transcript of GP1$PAP$S2$HW4$Solutions2 Skill22b2Day22 ...
GP1-‐PAP-‐S2-‐HW4-‐Solutions Skill 2b Day 2: Solving Quadratic Equations
Use the discriminant to state the number and types of solutions to the following equations. 1. 2𝑎! + 6𝑎 − 7 = 2 discriminant = 𝑏! − 4𝑎𝑐 = 108, which is positive and not a perfect square There are two distinct real irrational solutions. 2. 3𝑥! + 4𝑥 + 1 = 0 discriminant = 𝑏! − 4𝑎𝑐 = 4, which is positive and a perfect square There are two distinct real rational solutions. 3. 5𝑥! + 20𝑥 = 0 discriminant = 𝑏! − 4𝑎𝑐 = 400, which is positive and a perfect square There are two distinct real rational solutions. 4. 3𝑎! + 12𝑎 + 14 = 2 discriminant = 𝑏! − 4𝑎𝑐 = 0 There is exactly one real rational solution. 5. 5𝑦! + 2 = 4𝑦 discriminant = 𝑏! − 4𝑎𝑐 = −24, which is negative There are two distinct complex solutions (a complex conjugate pair). 6. 𝑐! + 6 = 0 discriminant = 𝑏! − 4𝑎𝑐 = −24, which is negative There are two distinct complex solutions (a complex conjugate pair). Determine the value of c that will complete the square. 7. 𝑥! − 14𝑥 + 𝑐
𝑐 =−142
!= 49
8. 𝑥! + 27𝑥 + 𝑐
𝑐 = !"!
!= !"#
!
GP1-‐PAP-‐S2-‐HW4-‐Solutions 9. 𝑥! − 5𝑥 + 𝑐
𝑐 =−52
!
=254
Solve the following quadratic equations by completing the square. Give exact answers. 10. 𝑥! − 14𝑥 + 40 = 0 𝑥! − 14𝑥 = −40 𝑥! − 14𝑥 + 49 = −40 + 49 𝑥 − 7 ! = 9 𝑥 − 7 = ±3 𝑥 = 4, 𝑥 = 10 11. 𝑥! − 6𝑥 = 15 𝑥! − 6𝑥 + 9 = 15 + 9 𝑥 − 3 ! = 24 𝑥 − 3 = ± 24 𝑥 − 3 = ± 4 ∙ 6 𝑥 = 3 ± 2 6 12. 2𝑥! + 8𝑥 = 10 𝑥! + 4𝑥 = 5 𝑥! + 4𝑥 + 4 = 5 + 4 𝑥 + 2 ! = 9 𝑥 + 2 = ±3 𝑥 = −5, 𝑥 = 1
GP1-‐PAP-‐S2-‐HW4-‐Solutions 13. 4𝑥! − 5𝑥 = −1 4𝑥!
4−5𝑥4=−14
𝑥! −54𝑥 =
−14
𝑥! −54𝑥 +
2564
=−14∙1616
+2564
𝑥 −58
!
=964
𝑥 −58= ±
38
𝑥 =14, 𝑥 = 1
14. 3𝑥! + 6𝑥 + 10 = 0 3𝑥! + 6𝑥 = −10 3𝑥!
3+6𝑥3=−103
𝑥! + 2𝑥 =−103
𝑥! + 2𝑥 + 1 =−103
+ 1 ∙33
𝑥 + 1 ! =−73
𝑥 + 1 = ±−73∙33
𝑥 = −1 ±213
𝑖
GP1-‐PAP-‐S2-‐HW4-‐Solutions 15. 4𝑥! − 12𝑥 + 9 = 0 4𝑥! − 12𝑥 = −9 4𝑥!
4−12𝑥4
=−94
𝑥! − 3𝑥 =−94
𝑥! − 3𝑥 +94=−94+94
𝑥 −32
!= 0
𝑥 =32
16. 16𝑥! + 10𝑥 − 73 = 8𝑥! 8𝑥! + 10𝑥 = 73 8𝑥!
8+10𝑥8
=738
𝑥! +54𝑥 =
738
𝑥! +54𝑥 +
2564
=738∙88+2564
𝑥 +58
!
=60964
𝑥 =−58±
6098
GP1-‐PAP-‐S2-‐HW4-‐Solutions 17. −8𝑥! − 2𝑥 = −3𝑥 − 10𝑥! + 27 2𝑥! + 𝑥 = 27 2𝑥!
2+𝑥2=272
𝑥! +𝑥2+116
=272∙88+116
𝑥 +14
!=21716
𝑥 =−14±
2174
18. 2𝑥! − 3𝑥 + 49 = −2 2𝑥! − 3𝑥 = −51 2𝑥!
2−3𝑥2=−512
𝑥! −32𝑥 +
916
=−512
∙88+916
𝑥 −34
!=−39916
𝑥 =34±
3994
𝑖
GP1-‐PAP-‐S2-‐HW4-‐Solutions 19. 8𝑥! + 6𝑥 = −7𝑥 − 77 8𝑥! + 13𝑥 = −77 8𝑥!
8+13𝑥8
=−778
𝑥! +138𝑥 =
−778
𝑥! +138𝑥 +
169256
=−778
∙3232
+169256
𝑥 +1316
!=−2295256
𝑥 =−1316
±3 25516
𝑖