I'll solve each quadratic equation using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Let me work through each equation step by step.
1) 3g² - 17g - 6 = 0
- a = 3, b = -17, c = -6
- Discriminant: b² - 4ac = (-17)² - 4(3)(-6) = 289 + 72 = 361
- √361 = 19
- g = (17 ± 19) / 6
- g = (17 + 19) / 6 = 36/6 = 6
- g = (17 - 19) / 6 = -2/6 = -1/3
2) 4r² - 23r + 33 = 0
- a = 4, b = -23, c = 33
- Discriminant: (-23)² - 4(4)(33) = 529 - 528 = 1
- √1 = 1
- r = (23 ± 1) / 8
- r = (23 + 1) / 8 = 24/8 = 3
- r = (23 - 1) / 8 = 22/8 = 11/4
3) 24x² + 22x - 7 = 0
- a = 24, b = 22, c = -7
- Discriminant: 22² - 4(24)(-7) = 484 + 672 = 1156
- √1156 = 34
- x = (-22 ± 34) / 48
- x = (-22 + 34) / 48 = 12/48 = 1/4
- x = (-22 - 34) / 48 = -56/48 = -7/6
4) 2h² - 7h + 8 = 0
- a = 2, b = -7, c = 8
- Discriminant: (-7)² - 4(2)(8) = 49 - 64 = -15
- Since discriminant is negative, there are no real solutions (complex solutions only)
5) 20b² + 11b = 4
First rewrite: 20b² + 11b - 4 = 0
- a = 20, b = 11, c = -4
- Discriminant: 11² - 4(20)(-4) = 121 + 320 = 441
- √441 = 21
- b = (-11 ± 21) / 40
- b = (-11 + 21) / 40 = 10/40 = 1/4
- b = (-11 - 21) / 40 = -32/40 = -4/5
6) 12g² + 41g + 35 = 0
- a = 12, b = 41, c = 35
- Discriminant: 41² - 4(12)(35) = 1681 - 1680 = 1
- 1 = 1
- g = (-41 ± 1) / 24
- g = (-41 + 1) / 24 = -40/24 = -5/3
- g = (-41 - 1) / 24 = -42/24 = -7/4
7) 5n² + 19n + 18 = 0
- a = 5, b = 19, c = 18
- Discriminant: 19² - 4(5)(18) = 361 - 360 = 1
- 1 = 1
- n = (-19 ± 1) / 10
- n = (-19 + 1) / 10 = -18/10 = -9/5
- n = (-19 - 1) / 10 = -20/10 = -2
8) 6n² - 17n = 14
First rewrite: 6n² - 17n - 14 = 0
- a = 6, b = -17, c = -14
- Discriminant: (-17)² - 4(6)(-14) = 289 + 336 = 625
- √625 = 25
- n = (17 ± 25) / 12
- n = (17 + 25) / 12 = 42/12 = 7/2
- n = (17 - 25) / 12 = -8/12 = -2/3
Final Answer:
1) g = 6 or g = -1/3
2) r = 3 or r = 11/4
3) x = 1/4 or x = -7/6
4) No real solutions (discriminant is negative)
5) b = 1/4 or b = -4/5
6) g = -5/3 or g = -7/4
7) n = -9/5 or n = -2
8) n = 7/2 or n = -2/3
Parent Tip: Review the logic above to help your child master the concept of quadratic equations worksheet with answers.
