We are given six quadratic equations to solve. All are in standard form:
ax² + bx + c = 0
We will solve each by
factoring, when possible, since they appear designed for that method.
---
1. 2x² – 2x – 24 = 0
Step 1: Factor out the GCF (Greatest Common Factor) first.
All terms divisible by 2:
→ 2(x² – x – 12) = 0
Now factor the trinomial inside:
We need two numbers that multiply to
–12 and add to
–1 →
–4 and 3
So:
x² – x – 12 = (x – 4)(x + 3)
Thus:
2(x – 4)(x + 3) = 0
Set each factor to zero:
- x – 4 = 0 →
x = 4
- x + 3 = 0 →
x = –3
✔ Solutions: x = 4, x = –3
---
2. x² + 10x + 16 = 0
Find two numbers that multiply to
16 and add to
10 →
8 and 2
So:
(x + 8)(x + 2) = 0
Set each factor to zero:
- x + 8 = 0 →
x = –8
- x + 2 = 0 →
x = –2
✔ Solutions: x = –8, x = –2
---
3. 4x² + 15x + 9 = 0
This is a bit trickier — we’ll use
AC method or trial factoring.
We need two numbers that multiply to
4×9 = 36 and add to
15 →
12 and 3
Split middle term:
→ 4x² + 12x + 3x + 9 = 0
Group:
→ (4x² + 12x) + (3x + 9) = 0
→ 4x(x + 3) + 3(x + 3) = 0
→ (4x + 3)(x + 3) = 0
Set each factor to zero:
- 4x + 3 = 0 → x =
–3/4
- x + 3 = 0 → x =
–3
✔ Solutions: x = –3/4, x = –3
---
4. x² – 13x + 42 = 0
Find two numbers that multiply to
42 and add to
–13 →
–6 and –7
So:
(x – 6)(x – 7) = 0
Set each factor to zero:
- x – 6 = 0 →
x = 6
- x – 7 = 0 →
x = 7
✔ Solutions: x = 6, x = 7
---
5. 6x² – x – 40 = 0
Use AC method: 6 × (–40) = –240
Need two numbers that multiply to
–240 and add to
–1
Try:
–16 and 15 → (–16)(15) = –240, –16 + 15 = –1
✔
Split middle term:
→ 6x² – 16x + 15x – 40 = 0
Group:
→ (6x² – 16x) + (15x – 40) = 0
→ 2x(3x – 8) + 5(3x – 8) = 0
→ (2x + 5)(3x – 8) = 0
Set each factor to zero:
- 2x + 5 = 0 → x =
–5/2
- 3x – 8 = 0 → x =
8/3
✔ Solutions: x = –5/2, x = 8/3
---
6. 2x² + 13x + 15 = 0
AC method: 2 × 15 = 30
Need two numbers that multiply to
30 and add to
13 →
10 and 3
Split middle term:
→ 2x² + 10x + 3x + 15 = 0
Group:
→ (2x² + 10x) + (3x + 15) = 0
→ 2x(x + 5) + 3(x + 5) = 0
→ (2x + 3)(x + 5) = 0
Set each factor to zero:
- 2x + 3 = 0 → x =
–3/2
- x + 5 = 0 → x =
–5
✔ Solutions: x = –3/2, x = –5
---
##
✔ Final Answers:
1.
2x² – 2x – 24 = 0 →
x = 4, x = –3
2.
x² + 10x + 16 = 0 →
x = –8, x = –2
3.
4x² + 15x + 9 = 0 →
x = –3/4, x = –3
4.
x² – 13x + 42 = 0 →
x = 6, x = 7
5.
6x² – x – 40 = 0 →
x = –5/2, x = 8/3
6.
2x² + 13x + 15 = 0 →
x = –3/2, x = –5
Let me know if you’d like to see any solved using the quadratic formula instead!
Parent Tip: Review the logic above to help your child master the concept of solving quadratic equation worksheet.
