KS3 and KS4 Factorising Quadratics Worksheets - Quadratic ... - Free Printable

The task involves factoring quadratic expressions. Factoring a quadratic expression means rewriting it as a product of two binomials (or other simpler factors). Here's how to approach the problem step by step:

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General Steps for Factoring Quadratic Expressions:

1. Identify the form: A quadratic expression is generally in the form \( ax^2 + bx + c \).
2. Look for common factors: Factor out any common factor from all terms.
3. Factor into binomials:
- For expressions of the form \( x^2 + bx + c \) (where \( a = 1 \)):
- Find two numbers that multiply to \( c \) and add to \( b \).
- For expressions of the form \( ax^2 + bx + c \) (where \( a \neq 1 \)):
- Use the "ac method" or trial and error to find the correct pair of binomials.
4. Check your work: Multiply the factors back to ensure you get the original quadratic expression.

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Solutions for Each Section

#### Section A
1. \( x^2 + 7x - 30 \)
- Find two numbers that multiply to \(-30\) and add to \(7\): \(10\) and \(-3\).
- Factor: \( (x + 10)(x - 3) \).

2. \( x^2 + 9x + 20 \)
- Find two numbers that multiply to \(20\) and add to \(9\): \(5\) and \(4\).
- Factor: \( (x + 5)(x + 4) \).

3. \( x^2 - 8x - 9 \)
- Find two numbers that multiply to \(-9\) and add to \(-8\): \(-9\) and \(1\).
- Factor: \( (x - 9)(x + 1) \).

4. \( x^2 + 8x + 7 \)
- Find two numbers that multiply to \(7\) and add to \(8\): \(7\) and \(1\).
- Factor: \( (x + 7)(x + 1) \).

5. \( x^2 - 9x - 22 \)
- Find two numbers that multiply to \(-22\) and add to \(-9\): \(-11\) and \(2\).
- Factor: \( (x - 11)(x + 2) \).

6. \( x^2 - x - 12 \)
- Find two numbers that multiply to \(-12\) and add to \(-1\): \(-4\) and \(3\).
- Factor: \( (x - 4)(x + 3) \).

7. \( x^2 - 17x + 72 \)
- Find two numbers that multiply to \(72\) and add to \(-17\): \(-8\) and \(-9\).
- Factor: \( (x - 8)(x - 9) \).

8. \( x^2 - x - 42 \)
- Find two numbers that multiply to \(-42\) and add to \(-1\): \(-7\) and \(6\).
- Factor: \( (x - 7)(x + 6) \).

9. \( x^2 - 13x + 56 \)
- Find two numbers that multiply to \(56\) and add to \(-13\): \(-7\) and \(-8\).
- Factor: \( (x - 7)(x - 8) \).

10. \( x^2 - 15x + 56 \)
- Find two numbers that multiply to \(56\) and add to \(-15\): \(-7\) and \(-8\).
- Factor: \( (x - 7)(x - 8) \).

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#### Section B
1. \( 2x^2 + 3x + 1 \)
- Use the "ac method":
- \( ac = 2 \cdot 1 = 2 \).
- Find two numbers that multiply to \(2\) and add to \(3\): \(2\) and \(1\).
- Rewrite: \( 2x^2 + 2x + x + 1 \).
- Group: \( (2x^2 + 2x) + (x + 1) \).
- Factor: \( 2x(x + 1) + 1(x + 1) \).
- Final factor: \( (2x + 1)(x + 1) \).

2. \( 2x^2 + 9x + 9 \)
- Use the "ac method":
- \( ac = 2 \cdot 9 = 18 \).
- Find two numbers that multiply to \(18\) and add to \(9\): \(6\) and \(3\).
- Rewrite: \( 2x^2 + 6x + 3x + 9 \).
- Group: \( (2x^2 + 6x) + (3x + 9) \).
- Factor: \( 2x(x + 3) + 3(x + 3) \).
- Final factor: \( (2x + 3)(x + 3) \).

3. \( 2x^2 + 5x + 2 \)
- Use the "ac method":
- \( ac = 2 \cdot 2 = 4 \).
- Find two numbers that multiply to \(4\) and add to \(5\): \(4\) and \(1\).
- Rewrite: \( 2x^2 + 4x + x + 2 \).
- Group: \( (2x^2 + 4x) + (x + 2) \).
- Factor: \( 2x(x + 2) + 1(x + 2) \).
- Final factor: \( (2x + 1)(x + 2) \).

4. \( 2x^2 + 7x + 5 \)
- Use the "ac method":
- \( ac = 2 \cdot 5 = 10 \).
- Find two numbers that multiply to \(10\) and add to \(7\): \(5\) and \(2\).
- Rewrite: \( 2x^2 + 5x + 2x + 5 \).
- Group: \( (2x^2 + 5x) + (2x + 5) \).
- Factor: \( x(2x + 5) + 1(2x + 5) \).
- Final factor: \( (2x + 5)(x + 1) \).

5. \( 2x^2 + 9x + 3 \)
- Use the "ac method":
- \( ac = 2 \cdot 3 = 6 \).
- Find two numbers that multiply to \(6\) and add to \(9\): \(6\) and \(3\).
- Rewrite: \( 2x^2 + 6x + 3x + 3 \).
- Group: \( (2x^2 + 6x) + (3x + 3) \).
- Factor: \( 2x(x + 3) + 3(x + 1) \).
- Final factor: \( (2x + 3)(x + 1) \).

6. \( 2x^2 + 18x + 14 \)
- Factor out the GCF (\(2\)): \( 2(x^2 + 9x + 7) \).
- Use the "ac method" on \( x^2 + 9x + 7 \):
- \( ac = 1 \cdot 7 = 7 \).
- Find two numbers that multiply to \(7\) and add to \(9\): \(7\) and \(2\).
- Rewrite: \( x^2 + 7x + 2x + 7 \).
- Group: \( (x^2 + 7x) + (2x + 7) \).
- Factor: \( x(x + 7) + 1(x + 7) \).
- Final factor: \( 2(x + 7)(x + 1) \).

7. \( 2x^2 + 22x + 36 \)
- Factor out the GCF (\(2\)): \( 2(x^2 + 11x + 18) \).
- Use the "ac method" on \( x^2 + 11x + 18 \):
- \( ac = 1 \cdot 18 = 18 \).
- Find two numbers that multiply to \(18\) and add to \(11\): \(9\) and \(2\).
- Rewrite: \( x^2 + 9x + 2x + 18 \).
- Group: \( (x^2 + 9x) + (2x + 18) \).
- Factor: \( x(x + 9) + 2(x + 9) \).
- Final factor: \( 2(x + 9)(x + 2) \).

8. \( 2x^2 + 29x + 48 \)
- Use the "ac method":
- \( ac = 2 \cdot 48 = 96 \).
- Find two numbers that multiply to \(96\) and add to \(29\): \(16\) and \(13\).
- Rewrite: \( 2x^2 + 16x + 13x + 48 \).
- Group: \( (2x^2 + 16x) + (13x + 48) \).
- Factor: \( 2x(x + 8) + 3(x + 8) \).
- Final factor: \( (2x + 3)(x + 8) \).

9. \( 2x^2 + 25x + 18 \)
- Use the "ac method":
- \( ac = 2 \cdot 18 = 36 \).
- Find two numbers that multiply to \(36\) and add to \(25\): \(18\) and \(7\).
- Rewrite: \( 2x^2 + 18x + 7x + 18 \).
- Group: \( (2x^2 + 18x) + (7x + 18) \).
- Factor: \( 2x(x + 9) + 1(x + 9) \).
- Final factor: \( (2x + 1)(x + 9) \).

10. \( 2x^2 + 23x + 12 \)
- Use the "ac method":
- \( ac = 2 \cdot 12 = 24 \).
- Find two numbers that multiply to \(24\) and add to \(23\): \(24\) and \(1\).
- Rewrite: \( 2x^2 + 24x + x + 12 \).
- Group: \( (2x^2 + 24x) + (x + 12) \).
- Factor: \( 2x(x + 12) + 1(x + 12) \).
- Final factor: \( (2x + 1)(x + 12) \).

11. \( 2x^2 + 13x + 18 \)
- Use the "ac method":
- \( ac = 2 \cdot 18 = 36 \).
- Find two numbers that multiply to \(36\) and add to \(13\): \(9\) and \(4\).
- Rewrite: \( 2x^2 + 9x + 4x + 18 \).
- Group: \( (2x^2 + 9x) + (4x + 18) \).
- Factor: \( x(2x + 9) + 2(2x + 9) \).
- Final factor: \( (2x + 9)(x + 2) \).

12. \( 2x^2 + 19x + 20 \)
- Use the "ac method":
- \( ac = 2 \cdot 20 = 40 \).
- Find two numbers that multiply to \(40\) and add to \(19\): \(16\) and \(3\).
- Rewrite: \( 2x^2 + 16x + 3x + 20 \).
- Group: \( (2x^2 + 16x) + (3x + 20) \).
- Factor: \( 2x(x + 8) + 1(x + 8) \).
- Final factor: \( (2x + 1)(x + 8) \).

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Final Answer

The solutions are boxed below for each section:

Section A:
1. \( (x + 10)(x - 3) \)
2. \( (x + 5)(x + 4) \)
3. \( (x - 9)(x + 1) \)
4. \( (x + 7)(x + 1) \)
5. \( (x - 11)(x + 2) \)
6. \( (x - 4)(x + 3) \)
7. \( (x - 8)(x - 9) \)
8. \( (x - 7)(x + 6) \)
9. \( (x - 7)(x - 8) \)
10. \( (x - 7)(x - 8) \)

Section B:
1. \( (2x + 1)(x + 1) \)
2. \( (2x + 3)(x + 3) \)
3. \( (2x + 1)(x + 2) \)
4. \( (2x + 5)(x + 1) \)
5. \( (2x + 3)(x + 1) \)
6. \( 2(x + 7)(x + 1) \)
7. \( 2(x + 9)(x + 2) \)
8. \( (2x + 3)(x + 8) \)
9. \( (2x + 1)(x + 9) \)
10. \( (2x + 1)(x + 12) \)
11. \( (2x + 9)(x + 2) \)
12. \( (2x + 1)(x + 8) \)

Final Boxed Answer:
\[
\boxed{
\begin{aligned}
&\text{Section A:} \\
&1. (x + 10)(x - 3) \\
&2. (x + 5)(x + 4) \\
&3. (x - 9)(x + 1) \\
&4. (x + 7)(x + 1) \\
&5. (x - 11)(x + 2) \\
&6. (x - 4)(x + 3) \\
&7. (x - 8)(x - 9) \\
&8. (x - 7)(x + 6) \\
&9. (x - 7)(x - 8) \\
&10. (x - 7)(x - 8) \\
\\
&\text{Section B:} \\
&1. (2x + 1)(x + 1) \\
&2. (2x + 3)(x + 3) \\
&3. (2x + 1)(x + 2) \\
&4. (2x + 5)(x + 1) \\
&5. (2x + 3)(x + 1) \\
&6. 2(x + 7)(x + 1) \\
&7. 2(x + 9)(x + 2) \\
&8. (2x + 3)(x + 8) \\
&9. (2x + 1)(x + 9) \\
&10. (2x + 1)(x + 12) \\
&11. (2x + 9)(x + 2) \\
&12. (2x + 1)(x + 8) \\
\end{aligned}
}
\]