Solving Quadratic Equations by Factoring worksheet - Free Printable

We are given six quadratic equations to solve. All are in standard form: `ax² + bx + c = 0`. We can solve them by factoring, if possible, or using the quadratic formula.

Let’s solve each one step-by-step.

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1. 2x² – 2x – 24 = 0

Step 1: Simplify by dividing all terms by 2

→ x² – x – 12 = 0

Step 2: Factor

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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2. x² + 10x + 16 = 0

Factor: 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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3. 4x² + 15x + 9 = 0

This is a bit trickier. Try factoring by grouping or AC method.

Multiply a×c = 4×9 = 36

Find 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

→ 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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4. x² – 13x + 42 = 0

Factor: 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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5. 6x² – x – 40 = 0

AC method: a×c = 6×(-40) = -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

→ 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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6. 2x² + 13x + 15 = 0

AC method: a×c = 2×15 = 30

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

→ 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 = -3/4, 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

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💡 Tip: Always check your solutions by plugging them back into the original equation to verify they satisfy it!

Let me know if you want to see any solved using the quadratic formula instead!