Let’s solve each quadratic equation using the quadratic formula:
The quadratic formula is:
x = [-b ± √(b² - 4ac)] / (2a)
for any equation in the form
ax² + bx + c = 0
We’ll go one by one.
---
1. 3g² - 17g - 6 = 0
Here, a = 3, b = -17, c = -6
Discriminant D = b² - 4ac = (-17)² - 4(3)(-6) = 289 + 72 = 361
√361 = 19
g = [17 ± 19] / (2×3) = [17 ± 19]/6
→ g = (17 + 19)/6 = 36/6 = 6
→ g = (17 - 19)/6 = -2/6 = -1/3
✔ Solutions:
g = 6 or g = -1/3
---
2. 4r² - 23r + 33 = 0
a = 4, b = -23, c = 33
D = (-23)² - 4(4)(33) = 529 - 528 = 1
√1 = 1
r = [23 ± 1] / (2×4) = [23 ± 1]/8
→ r = 24/8 = 3
→ r = 22/8 = 11/4
✔ Solutions:
r = 3 or r = 11/4
---
3. 24x² + 22x - 7 = 0
a = 24, b = 22, c = -7
D = (22)² - 4(24)(-7) = 484 + 672 = 1156
√1156 = 34
x = [-22 ± 34] / (2×24) = [-22 ± 34]/48
→ x = (12)/48 = 1/4
→ x = (-56)/48 = -7/6
✔ Solutions:
x = 1/4 or x = -7/6
---
4. 2h² - 7h + 8 = 0
a = 2, b = -7, c = 8
D = (-7)² - 4(2)(8) = 49 - 64 = -15 → negative!
No real solutions — complex numbers only.
But since this is likely for real-number context, we say:
✔ No real solutions.
*(If needed: h = [7 ± √(-15)] / 4 = [7 ± i√15]/4)*
---
5. 20b² + 11b = 4 → rewrite as 20b² + 11b - 4 = 0
a = 20, b = 11, c = -4
D = (11)² - 4(20)(-4) = 121 + 320 = 441
√441 = 21
b = [-11 ± 21] / (2×20) = [-11 ± 21]/40
→ b = 10/40 = 1/4
→ b = -32/40 = -4/5
✔ Solutions:
b = 1/4 or b = -4/5
---
6. 12g² + 41g + 35 = 0
a = 12, b = 41, c = 35
D = (41)² - 4(12)(35) = 1681 - 1680 = 1
√1 = 1
g = [-41 ± 1] / (2×12) = [-41 ± 1]/24
→ g = -40/24 = -5/3
→ g = -42/24 = -7/4
✔ Solutions:
g = -5/3 or g = -7/4
---
7. 5n² + 19n + 18 = 0
a = 5, b = 19, c = 18
D = (19)² - 4(5)(18) = 361 - 360 = 1
√1 = 1
n = [-19 ± 1] / (2×5) = [-19 ± 1]/10
→ n = -18/10 = -9/5
→ n = -20/10 = -2
✔ Solutions:
n = -9/5 or n = -2
---
8. 6n² - 17n = 14 → rewrite as 6n² - 17n - 14 = 0
a = 6, b = -17, c = -14
D = (-17)² - 4(6)(-14) = 289 + 336 = 625
√625 = 25
n = [17 ± 25] / (2×6) = [17 ± 25]/12
→ n = 42/12 = 7/2
→ n = -8/12 = -2/3
✔ Solutions:
n = 7/2 or n = -2/3
---
Final Answer:
1. g = 6, -1/3
2. r = 3, 11/4
3. x = 1/4, -7/6
4. No real solutions
5. b = 1/4, -4/5
6. g = -5/3, -7/4
7. n = -9/5, -2
8. n = 7/2, -2/3
Parent Tip: Review the logic above to help your child master the concept of solving quadratic equations worksheet with answers.
