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GP1-‐PAP-‐S2-‐HW4-‐Answers 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 pure imaginary 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-‐Answers 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 − 7 ! = 9 𝑥 = 4, 𝑥 = 10 11. 𝑥! − 6𝑥 = 15 𝑥 − 3 ! = 24 𝑥 = 3 ± 2 6 12. 2𝑥! + 8𝑥 = 10 𝑥 + 2 ! = 9 𝑥 = −5, 𝑥 = 1 13. 4𝑥! − 5𝑥 = −1
𝑥 −58
!
=964
𝑥 =14, 𝑥 = 1
14. 3𝑥! + 6𝑥 + 10 = 0
𝑥 + 1 ! =−73
𝑥 = −1 ±213
𝑖 15. 4𝑥! − 12𝑥 + 9 = 0
𝑥 −32
!= 0
𝑥 =32