Factoring Quadratic Expressions worksheet with 20 problems for practice.

The task is to factor each quadratic expression. Factoring a quadratic expression involves rewriting it as a product of two binomials (or other simpler factors). Here's how we can solve each problem step by step.

General Approach:

1. Identify the form: The general form of a quadratic expression is \( ax^2 + bx + c \).
2. Find factors of \( ac \): Look for two numbers that multiply to \( ac \) and add up to \( b \).
3. Rewrite the middle term: Split the middle term using the two numbers found.
4. Factor by grouping: Group the terms and factor out common factors.

Let’s solve each problem:

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Problem 1: \( x^2 - 4x - 45 \)

- Step 1: Identify \( a = 1 \), \( b = -4 \), \( c = -45 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-45) = -45 \) and add up to \( b = -4 \).
- The numbers are \( -9 \) and \( 5 \) because \( -9 \cdot 5 = -45 \) and \( -9 + 5 = -4 \).
- Step 3: Rewrite the middle term:
\[
x^2 - 4x - 45 = x^2 - 9x + 5x - 45
\]
- Step 4: Factor by grouping:
\[
x^2 - 9x + 5x - 45 = x(x - 9) + 5(x - 9) = (x - 9)(x + 5)
\]

Answer: \( (x - 9)(x + 5) \)

---

Problem 2: \( x^2 - 5x - 6 \)

- Step 1: Identify \( a = 1 \), \( b = -5 \), \( c = -6 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-6) = -6 \) and add up to \( b = -5 \).
- The numbers are \( -6 \) and \( 1 \) because \( -6 \cdot 1 = -6 \) and \( -6 + 1 = -5 \).
- Step 3: Rewrite the middle term:
\[
x^2 - 5x - 6 = x^2 - 6x + x - 6
\]
- Step 4: Factor by grouping:
\[
x^2 - 6x + x - 6 = x(x - 6) + 1(x - 6) = (x - 6)(x + 1)
\]

Answer: \( (x - 6)(x + 1) \)

---

Problem 3: \( x^2 - 17x + 72 \)

- Step 1: Identify \( a = 1 \), \( b = -17 \), \( c = 72 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot 72 = 72 \) and add up to \( b = -17 \).
- The numbers are \( -8 \) and \( -9 \) because \( -8 \cdot -9 = 72 \) and \( -8 + (-9) = -17 \).
- Step 3: Rewrite the middle term:
\[
x^2 - 17x + 72 = x^2 - 8x - 9x + 72
\]
- Step 4: Factor by grouping:
\[
x^2 - 8x - 9x + 72 = x(x - 8) - 9(x - 8) = (x - 8)(x - 9)
\]

Answer: \( (x - 8)(x - 9) \)

---

Problem 4: \( x^2 - 36 \)

- Step 1: Recognize this as a difference of squares:
\[
x^2 - 36 = x^2 - 6^2
\]
- Step 2: Use the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \):
\[
x^2 - 6^2 = (x - 6)(x + 6)
\]

Answer: \( (x - 6)(x + 6) \)

---

Problem 5: \( x^2 + 7x - 8 \)

- Step 1: Identify \( a = 1 \), \( b = 7 \), \( c = -8 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-8) = -8 \) and add up to \( b = 7 \).
- The numbers are \( 8 \) and \( -1 \) because \( 8 \cdot (-1) = -8 \) and \( 8 + (-1) = 7 \).
- Step 3: Rewrite the middle term:
\[
x^2 + 7x - 8 = x^2 + 8x - x - 8
\]
- Step 4: Factor by grouping:
\[
x^2 + 8x - x - 8 = x(x + 8) - 1(x + 8) = (x + 8)(x - 1)
\]

Answer: \( (x + 8)(x - 1) \)

---

Problem 6: \( x^2 - 3x - 54 \)

- Step 1: Identify \( a = 1 \), \( b = -3 \), \( c = -54 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-54) = -54 \) and add up to \( b = -3 \).
- The numbers are \( -9 \) and \( 6 \) because \( -9 \cdot 6 = -54 \) and \( -9 + 6 = -3 \).
- Step 3: Rewrite the middle term:
\[
x^2 - 3x - 54 = x^2 - 9x + 6x - 54
\]
- Step 4: Factor by grouping:
\[
x^2 - 9x + 6x - 54 = x(x - 9) + 6(x - 9) = (x - 9)(x + 6)
\]

Answer: \( (x - 9)(x + 6) \)

---

Problem 7: \( x^2 + 3x + 2 \)

- Step 1: Identify \( a = 1 \), \( b = 3 \), \( c = 2 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot 2 = 2 \) and add up to \( b = 3 \).
- The numbers are \( 2 \) and \( 1 \) because \( 2 \cdot 1 = 2 \) and \( 2 + 1 = 3 \).
- Step 3: Rewrite the middle term:
\[
x^2 + 3x + 2 = x^2 + 2x + x + 2
\]
- Step 4: Factor by grouping:
\[
x^2 + 2x + x + 2 = x(x + 2) + 1(x + 2) = (x + 2)(x + 1)
\]

Answer: \( (x + 2)(x + 1) \)

---

Problem 8: \( x^2 + 3x - 18 \)

- Step 1: Identify \( a = 1 \), \( b = 3 \), \( c = -18 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-18) = -18 \) and add up to \( b = 3 \).
- The numbers are \( 6 \) and \( -3 \) because \( 6 \cdot (-3) = -18 \) and \( 6 + (-3) = 3 \).
- Step 3: Rewrite the middle term:
\[
x^2 + 3x - 18 = x^2 + 6x - 3x - 18
\]
- Step 4: Factor by grouping:
\[
x^2 + 6x - 3x - 18 = x(x + 6) - 3(x + 6) = (x + 6)(x - 3)
\]

Answer: \( (x + 6)(x - 3) \)

---

Problem 9: \( x^2 + 17x + 72 \)

- Step 1: Identify \( a = 1 \), \( b = 17 \), \( c = 72 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot 72 = 72 \) and add up to \( b = 17 \).
- The numbers are \( 8 \) and \( 9 \) because \( 8 \cdot 9 = 72 \) and \( 8 + 9 = 17 \).
- Step 3: Rewrite the middle term:
\[
x^2 + 17x + 72 = x^2 + 8x + 9x + 72
\]
- Step 4: Factor by grouping:
\[
x^2 + 8x + 9x + 72 = x(x + 8) + 9(x + 8) = (x + 8)(x + 9)
\]

Answer: \( (x + 8)(x + 9) \)

---

Problem 10: \( x^2 + 10x + 24 \)

- Step 1: Identify \( a = 1 \), \( b = 10 \), \( c = 24 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot 24 = 24 \) and add up to \( b = 10 \).
- The numbers are \( 6 \) and \( 4 \) because \( 6 \cdot 4 = 24 \) and \( 6 + 4 = 10 \).
- Step 3: Rewrite the middle term:
\[
x^2 + 10x + 24 = x^2 + 6x + 4x + 24
\]
- Step 4: Factor by grouping:
\[
x^2 + 6x + 4x + 24 = x(x + 6) + 4(x + 6) = (x + 6)(x + 4)
\]

Answer: \( (x + 6)(x + 4) \)

---

Problem 11: \( x^2 - 9x + 8 \)

- Step 1: Identify \( a = 1 \), \( b = -9 \), \( c = 8 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot 8 = 8 \) and add up to \( b = -9 \).
- The numbers are \( -8 \) and \( -1 \) because \( -8 \cdot (-1) = 8 \) and \( -8 + (-1) = -9 \).
- Step 3: Rewrite the middle term:
\[
x^2 - 9x + 8 = x^2 - 8x - x + 8
\]
- Step 4: Factor by grouping:
\[
x^2 - 8x - x + 8 = x(x - 8) - 1(x - 8) = (x - 8)(x - 1)
\]

Answer: \( (x - 8)(x - 1) \)

---

Problem 12: \( x^2 + x - 42 \)

- Step 1: Identify \( a = 1 \), \( b = 1 \), \( c = -42 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-42) = -42 \) and add up to \( b = 1 \).
- The numbers are \( 7 \) and \( -6 \) because \( 7 \cdot (-6) = -42 \) and \( 7 + (-6) = 1 \).
- Step 3: Rewrite the middle term:
\[
x^2 + x - 42 = x^2 + 7x - 6x - 42
\]
- Step 4: Factor by grouping:
\[
x^2 + 7x - 6x - 42 = x(x + 7) - 6(x + 7) = (x + 7)(x - 6)
\]

Answer: \( (x + 7)(x - 6) \)

---

Problem 13: \( x^2 - x - 72 \)

- Step 1: Identify \( a = 1 \), \( b = -1 \), \( c = -72 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-72) = -72 \) and add up to \( b = -1 \).
- The numbers are \( -9 \) and \( 8 \) because \( -9 \cdot 8 = -72 \) and \( -9 + 8 = -1 \).
- Step 3: Rewrite the middle term:
\[
x^2 - x - 72 = x^2 - 9x + 8x - 72
\]
- Step 4: Factor by grouping:
\[
x^2 - 9x + 8x - 72 = x(x - 9) + 8(x - 9) = (x - 9)(x + 8)
\]

Answer: \( (x - 9)(x + 8) \)

---

Problem 14: \( x^2 + 2x - 63 \)

- Step 1: Identify \( a = 1 \), \( b = 2 \), \( c = -63 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-63) = -63 \) and add up to \( b = 2 \).
- The numbers are \( 9 \) and \( -7 \) because \( 9 \cdot (-7) = -63 \) and \( 9 + (-7) = 2 \).
- Step 3: Rewrite the middle term:
\[
x^2 + 2x - 63 = x^2 + 9x - 7x - 63
\]
- Step 4: Factor by grouping:
\[
x^2 + 9x - 7x - 63 = x(x + 9) - 7(x + 9) = (x + 9)(x - 7)
\]

Answer: \( (x + 9)(x - 7) \)

---

Problem 15: \( x^2 + 13x + 40 \)

- Step 1: Identify \( a = 1 \), \( b = 13 \), \( c = 40 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot 40 = 40 \) and add up to \( b = 13 \).
- The numbers are \( 8 \) and \( 5 \) because \( 8 \cdot 5 = 40 \) and \( 8 + 5 = 13 \).
- Step 3: Rewrite the middle term:
\[
x^2 + 13x + 40 = x^2 + 8x + 5x + 40
\]
- Step 4: Factor by grouping:
\[
x^2 + 8x + 5x + 40 = x(x + 8) + 5(x + 8) = (x + 8)(x + 5)
\]

Answer: \( (x + 8)(x + 5) \)

---

Problem 16: \( x^2 - 2x - 8 \)

- Step 1: Identify \( a = 1 \), \( b = -2 \), \( c = -8 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-8) = -8 \) and add up to \( b = -2 \).
- The numbers are \( -4 \) and \( 2 \) because \( -4 \cdot 2 = -8 \) and \( -4 + 2 = -2 \).
- Step 3: Rewrite the middle term:
\[
x^2 - 2x - 8 = x^2 - 4x + 2x - 8
\]
- Step 4: Factor by grouping:
\[
x^2 - 4x + 2x - 8 = x(x - 4) + 2(x - 4) = (x - 4)(x + 2)
\]

Answer: \( (x - 4)(x + 2) \)

---

Problem 17: \( x^2 + x - 6 \)

- Step 1: Identify \( a = 1 \), \( b = 1 \), \( c = -6 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-6) = -6 \) and add up to \( b = 1 \).
- The numbers are \( 3 \) and \( -2 \) because \( 3 \cdot (-2) = -6 \) and \( 3 + (-2) = 1 \).
- Step 3: Rewrite the middle term:
\[
x^2 + x - 6 = x^2 + 3x - 2x - 6
\]
- Step 4: Factor by grouping:
\[
x^2 + 3x - 2x - 6 = x(x + 3) - 2(x + 3) = (x + 3)(x - 2)
\]

Answer: \( (x + 3)(x - 2) \)

---

Problem 18: \( x^2 - 4x + 3 \)

- Step 1: Identify \( a = 1 \), \( b = -4 \), \( c = 3 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot 3 = 3 \) and add up to \( b = -4 \).
- The numbers are \( -3 \) and \( -1 \) because \( -3 \cdot (-1) = 3 \) and \( -3 + (-1) = -4 \).
- Step 3: Rewrite the middle term:
\[
x^2 - 4x + 3 = x^2 - 3x - x + 3
\]
- Step 4: Factor by grouping:
\[
x^2 - 3x - x + 3 = x(x - 3) - 1(x - 3) = (x - 3)(x - 1)
\]

Answer: \( (x - 3)(x - 1) \)

---

Problem 19: \( x^2 + 4x - 5 \)

- Step 1: Identify \( a = 1 \), \( b = 4 \), \( c = -5 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-5) = -5 \) and add up to \( b = 4 \).
- The numbers are \( 5 \) and \( -1 \) because \( 5 \cdot (-1) = -5 \) and \( 5 + (-1) = 4 \).
- Step 3: Rewrite the middle term:
\[
x^2 + 4x - 5 = x^2 + 5x - x - 5
\]
- Step 4: Factor by grouping:
\[
x^2 + 5x - x - 5 = x(x + 5) - 1(x + 5) = (x + 5)(x - 1)
\]

Answer: \( (x + 5)(x - 1) \)

---

Problem 20: \( x^2 - 2x - 35 \)

- Step 1: Identify \( a = 1 \), \( b = -2 \), \( c = -35 \).
- Step 2: Find two numbers that multiply to \( ac = 1 \cdot (-35) = -35 \) and add up to \( b = -2 \).
- The numbers are \( -7 \) and \( 5 \) because \( -7 \cdot 5 = -35 \) and \( -7 + 5 = -2 \).
- Step 3: Rewrite the middle term:
\[
x^2 - 2x - 35 = x^2 - 7x + 5x - 35
\]
- Step 4: Factor by grouping:
\[
x^2 - 7x + 5x - 35 = x(x - 7) + 5(x - 7) = (x - 7)(x + 5)
\]

Answer: \( (x - 7)(x + 5) \)

---

Final Answer:

\[
\boxed{
\begin{aligned}
1. & \ (x - 9)(x + 5) \\
2. & \ (x - 6)(x + 1) \\
3. & \ (x - 8)(x - 9) \\
4. & \ (x - 6)(x + 6) \\
5. & \ (x + 8)(x - 1) \\
6. & \ (x - 9)(x + 6) \\
7. & \ (x + 2)(x + 1) \\
8. & \ (x + 6)(x - 3) \\
9. & \ (x + 8)(x + 9) \\
10. & \ (x + 6)(x + 4) \\
11. & \ (x - 8)(x - 1) \\
12. & \ (x + 7)(x - 6) \\
13. & \ (x - 9)(x + 8) \\
14. & \ (x + 9)(x - 7) \\
15. & \ (x + 8)(x + 5) \\
16. & \ (x - 4)(x + 2) \\
17. & \ (x + 3)(x - 2) \\
18. & \ (x - 3)(x - 1) \\
19. & \ (x + 5)(x - 1) \\
20. & \ (x - 7)(x + 5)
\end{aligned}
}
\]