Solving Quadratic Equations by Factoring worksheet - Free Printable

We are given six quadratic equations to solve. All of them are in standard form:
ax² + bx + c = 0

We’ll solve each using factoring, since they appear designed for that method (integer solutions likely). If factoring is not obvious, we can use the quadratic formula, but let’s try factoring first.

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🔹 Equation 1:

2x² - 2x - 24 = 0

✔ Step 1: Factor out the GCF (Greatest Common Factor)
All coefficients divisible by 2:
→ 2(x² - x - 12) = 0
→ x² - x - 12 = 0

✔ Step 2: Factor the trinomial
We need two numbers that multiply to -12 and add to -1 → -4 and +3
→ (x - 4)(x + 3) = 0

✔ Step 3: Solve
x - 4 = 0 → x = 4
x + 3 = 0 → x = -3

✔ Solutions: x = 4, x = -3

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🔹 Equation 2:

x² + 10x + 16 = 0

✔ Find two numbers that multiply to 16 and add to 10 → 8 and 2
→ (x + 8)(x + 2) = 0

✔ Solve:
x + 8 = 0 → x = -8
x + 2 = 0 → x = -2

✔ Solutions: x = -8, x = -2

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🔹 Equation 3:

4x² + 15x + 9 = 0

✔ This is a bit trickier. Use AC method or trial factoring.
a = 4, c = 9 → ac = 36
Need two numbers that multiply to 36 and add to 15 → 12 and 3

✔ Rewrite middle term:
4x² + 12x + 3x + 9 = 0
Group: (4x² + 12x) + (3x + 9) = 0
Factor: 4x(x + 3) + 3(x + 3) = 0
→ (4x + 3)(x + 3) = 0

✔ Solve:
4x + 3 = 0 → x = -3/4
x + 3 = 0 → x = -3

✔ Solutions: x = -3/4, x = -3

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🔹 Equation 4:

x² - 13x + 42 = 0

✔ Two numbers that multiply to 42 and add to -13 → -6 and -7
→ (x - 6)(x - 7) = 0

✔ Solve:
x - 6 = 0 → x = 6
x - 7 = 0 → x = 7

✔ Solutions: x = 6, x = 7

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🔹 Equation 5:

6x² - x - 40 = 0

✔ AC method: a=6, c=-40 → ac = -240
Need two numbers that multiply to -240 and add to -1 → -16 and 15

✔ Rewrite:
6x² - 16x + 15x - 40 = 0
Group: (6x² - 16x) + (15x - 40) = 0
Factor: 2x(3x - 8) + 5(3x - 8) = 0
→ (2x + 5)(3x - 8) = 0

✔ Solve:
2x + 5 = 0 → x = -5/2
3x - 8 = 0 → x = 8/3

✔ Solutions: x = -5/2, x = 8/3

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🔹 Equation 6:

2x² + 13x + 15 = 0

✔ AC method: a=2, c=15 → ac=30
Need two numbers that multiply to 30 and add to 13 → 10 and 3

✔ Rewrite:
2x² + 10x + 3x + 15 = 0
Group: (2x² + 10x) + (3x + 15) = 0
Factor: 2x(x + 5) + 3(x + 5) = 0
→ (2x + 3)(x + 5) = 0

✔ Solve:
2x + 3 = 0 → x = -3/2
x + 5 = 0 → x = -5

✔ Solutions: x = -3/2, x = -5

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## ✔ 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, x = -3/4
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 = -5, x = -3/2

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💡 Note: All equations were solved by factoring (with some requiring grouping or AC method). You could also use the quadratic formula for verification, but factoring is efficient here since all have rational roots.

Let me know if you want step-by-step verification using the quadratic formula!