Factoring Quadratic Expressions Worksheet | Fun and Engaging ... - Free Printable

Problem: Factoring Quadratic Expressions

The task involves simplifying and factoring quadratic expressions. Let's solve the problems step by step.

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#### Section A: Simplify then factorise the following quadratic expressions.

1. \( x^2 - 6x - 2x + 12 \)
- Simplify: \( x^2 - 8x + 12 \)
- Factorise: Look for two numbers that multiply to \( 12 \) and add to \( -8 \). These numbers are \( -6 \) and \( -2 \).
\[
x^2 - 8x + 12 = (x - 6)(x - 2)
\]

2. \( d(d - 5) - 84 \)
- Expand: \( d^2 - 5d - 84 \)
- Factorise: Look for two numbers that multiply to \( -84 \) and add to \( -5 \). These numbers are \( -12 \) and \( 7 \).
\[
d^2 - 5d - 84 = (d - 12)(d + 7)
\]

3. \( b^2 + 2(b - 4) \)
- Expand: \( b^2 + 2b - 8 \)
- Factorise: Look for two numbers that multiply to \( -8 \) and add to \( 2 \). These numbers are \( 4 \) and \( -2 \).
\[
b^2 + 2b - 8 = (b + 4)(b - 2)
\]

4. \( x^2 - 3(2x + 9) \)
- Expand: \( x^2 - 6x - 27 \)
- Factorise: Look for two numbers that multiply to \( -27 \) and add to \( -6 \). These numbers are \( -9 \) and \( 3 \).
\[
x^2 - 6x - 27 = (x - 9)(x + 3)
\]

5. \( c(c + 8) - 48 \)
- Expand: \( c^2 + 8c - 48 \)
- Factorise: Look for two numbers that multiply to \( -48 \) and add to \( 8 \). These numbers are \( 12 \) and \( -4 \).
\[
c^2 + 8c - 48 = (c + 12)(c - 4)
\]

6. \( 3a(a - 2) - 4a + 3 \)
- Expand: \( 3a^2 - 6a - 4a + 3 = 3a^2 - 10a + 3 \)
- Factorise: Look for two numbers that multiply to \( 3 \times 3 = 9 \) and add to \( -10 \). These numbers are \( -9 \) and \( -1 \).
\[
3a^2 - 10a + 3 = (3a - 1)(a - 3)
\]

7. \( 5w(w - 2) - 4w - 3 \)
- Expand: \( 5w^2 - 10w - 4w - 3 = 5w^2 - 14w - 3 \)
- Factorise: Look for two numbers that multiply to \( 5 \times -3 = -15 \) and add to \( -14 \). These numbers are \( -15 \) and \( 1 \).
\[
5w^2 - 14w - 3 = (5w + 1)(w - 3)
\]

8. \( 3(6 - 5s) + s^2 + s^2 \)
- Simplify: \( 3(6 - 5s) + 2s^2 = 18 - 15s + 2s^2 \)
- Rearrange: \( 2s^2 - 15s + 18 \)
- Factorise: Look for two numbers that multiply to \( 2 \times 18 = 36 \) and add to \( -15 \). These numbers are \( -12 \) and \( -3 \).
\[
2s^2 - 15s + 18 = (2s - 3)(s - 6)
\]

9. \( 3 + 2y(4y + 5) \)
- Expand: \( 3 + 8y^2 + 10y = 8y^2 + 10y + 3 \)
- Factorise: Look for two numbers that multiply to \( 8 \times 3 = 24 \) and add to \( 10 \). These numbers are \( 6 \) and \( 4 \).
\[
8y^2 + 10y + 3 = (2y + 3)(4y + 1)
\]

10. \( 9x^2 - (x - 3)^2 \)
- Expand: \( 9x^2 - (x^2 - 6x + 9) = 9x^2 - x^2 + 6x - 9 = 8x^2 + 6x - 9 \)
- Factorise: Look for two numbers that multiply to \( 8 \times -9 = -72 \) and add to \( 6 \). These numbers are \( 12 \) and \( -6 \).
\[
8x^2 + 6x - 9 = (2x + 3)(4x - 3)
\]

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#### Section B: Factorise the following quadratic expressions.

1. \( x^2 - 4 \)
- Difference of squares: \( x^2 - 4 = (x - 2)(x + 2) \)

2. \( s^2 - 25 \)
- Difference of squares: \( s^2 - 25 = (s - 5)(s + 5) \)

3. \( t^2 - 64 \)
- Difference of squares: \( t^2 - 64 = (t - 8)(t + 8) \)

4. \( 9 - y^2 \)
- Difference of squares: \( 9 - y^2 = (3 - y)(3 + y) \)

5. \( 49 - p^2 \)
- Difference of squares: \( 49 - p^2 = (7 - p)(7 + p) \)

6. \( 4q^2 - 121 \)
- Difference of squares: \( 4q^2 - 121 = (2q - 11)(2q + 11) \)

7. \( 81 - 25k^2 \)
- Difference of squares: \( 81 - 25k^2 = (9 - 5k)(9 + 5k) \)

8. \( 1 - 400d^2 \)
- Difference of squares: \( 1 - 400d^2 = (1 - 20d)(1 + 20d) \)

9. \( 600v^2 - 6 \)
- Factor out the GCF: \( 6(100v^2 - 1) \)
- Difference of squares: \( 100v^2 - 1 = (10v - 1)(10v + 1) \)
\[
600v^2 - 6 = 6(10v - 1)(10v + 1)
\]

10. \( a^2 - b^2 \)
- Difference of squares: \( a^2 - b^2 = (a - b)(a + b) \)

11. \( x^2 - 9y^2 \)
- Difference of squares: \( x^2 - 9y^2 = (x - 3y)(x + 3y) \)

12. \( 4c^2 - d^2 \)
- Difference of squares: \( 4c^2 - d^2 = (2c - d)(2c + d) \)

13. \( 16s^2 - 9t^2 \)
- Difference of squares: \( 16s^2 - 9t^2 = (4s - 3t)(4s + 3t) \)

14. \( 49w^2 - 100v^2 \)
- Difference of squares: \( 49w^2 - 100v^2 = (7w - 10v)(7w + 10v) \)

15. \( 32p^2 - 18q^2 \)
- Factor out the GCF: \( 2(16p^2 - 9q^2) \)
- Difference of squares: \( 16p^2 - 9q^2 = (4p - 3q)(4p + 3q) \)
\[
32p^2 - 18q^2 = 2(4p - 3q)(4p + 3q)
\]

16. \( 48x^2 - 12y^2 \)
- Factor out the GCF: \( 12(4x^2 - y^2) \)
- Difference of squares: \( 4x^2 - y^2 = (2x - y)(2x + y) \)
\[
48x^2 - 12y^2 = 12(2x - y)(2x + y)
\]

17. \( 45a^2 - 125b^2 \)
- Factor out the GCF: \( 5(9a^2 - 25b^2) \)
- Difference of squares: \( 9a^2 - 25b^2 = (3a - 5b)(3a + 5b) \)
\[
45a^2 - 125b^2 = 5(3a - 5b)(3a + 5b)
\]

18. \( 72x^2 - 242y^2 \)
- Factor out the GCF: \( 2(36x^2 - 121y^2) \)
- Difference of squares: \( 36x^2 - 121y^2 = (6x - 11y)(6x + 11y) \)
\[
72x^2 - 242y^2 = 2(6x - 11y)(6x + 11y)
\]

19. \( a^2b^2 - c^2 \)
- Difference of squares: \( a^2b^2 - c^2 = (ab - c)(ab + c) \)

20. \( 9s - 4s^3 \)
- Factor out the GCF: \( s(9 - 4s^2) \)
- Difference of squares: \( 9 - 4s^2 = (3 - 2s)(3 + 2s) \)
\[
9s - 4s^3 = s(3 - 2s)(3 + 2s)
\]

21. \( (xy)^2 - 4z^2 \)
- Difference of squares: \( (xy)^2 - 4z^2 = (xy - 2z)(xy + 2z) \)

22. \( 64t^4 - 16s^4 \)
- Factor out the GCF: \( 16(4t^4 - s^4) \)
- Difference of squares: \( 4t^4 - s^4 = (2t^2 - s^2)(2t^2 + s^2) \)
\[
64t^4 - 16s^4 = 16(2t^2 - s^2)(2t^2 + s^2)
\]

23. \( (4x^2)^2 - 36y^2 \)
- Difference of squares: \( (4x^2)^2 - 36y^2 = (4x^2 - 6y)(4x^2 + 6y) \)

24. \( 27a^4 - 12b^2 \)
- Factor out the GCF: \( 3(9a^4 - 4b^2) \)
- Difference of squares: \( 9a^4 - 4b^2 = (3a^2 - 2b)(3a^2 + 2b) \)
\[
27a^4 - 12b^2 = 3(3a^2 - 2b)(3a^2 + 2b)
\]

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#### Extension: Using the difference of two squares factorise the following expressions.

1. \( 4x^2 - (x - 2)^2 \)
- Difference of squares: \( 4x^2 - (x - 2)^2 = (2x - (x - 2))(2x + (x - 2)) \)
\[
4x^2 - (x - 2)^2 = (2x - x + 2)(2x + x - 2) = (x + 2)(3x - 2)
\]

2. \( (2x + 1)^2 - (x - 4)^2 \)
- Difference of squares: \( (2x + 1)^2 - (x - 4)^2 = ((2x + 1) - (x - 4))((2x + 1) + (x - 4)) \)
\[
(2x + 1)^2 - (x - 4)^2 = (2x + 1 - x + 4)(2x + 1 + x - 4) = (x + 5)(3x - 3) = 3(x + 5)(x - 1)
\]

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Final Answers:

\[
\boxed{
\begin{aligned}
&\text{Section A:} \\
&1. (x - 6)(x - 2) \\
&2. (d - 12)(d + 7) \\
&3. (b + 4)(b - 2) \\
&4. (x - 9)(x + 3) \\
&5. (c + 12)(c - 4) \\
&6. (3a - 1)(a - 3) \\
&7. (5w + 1)(w - 3) \\
&8. (2s - 3)(s - 6) \\
&9. (2y + 3)(4y + 1) \\
&10. (2x + 3)(4x - 3) \\
\\
&\text{Section B:} \\
&1. (x - 2)(x + 2) \\
&2. (s - 5)(s + 5) \\
&3. (t - 8)(t + 8) \\
&4. (3 - y)(3 + y) \\
&5. (7 - p)(7 + p) \\
&6. (2q - 11)(2q + 11) \\
&7. (9 - 5k)(9 + 5k) \\
&8. (1 - 20d)(1 + 20d) \\
&9. 6(10v - 1)(10v + 1) \\
&10. (a - b)(a + b) \\
&11. (x - 3y)(x + 3y) \\
&12. (2c - d)(2c + d) \\
&13. (4s - 3t)(4s + 3t) \\
&14. (7w - 10v)(7w + 10v) \\
&15. 2(4p - 3q)(4p + 3q) \\
&16. 12(2x - y)(2x + y) \\
&17. 5(3a - 5b)(3a + 5b) \\
&18. 2(6x - 11y)(6x + 11y) \\
&19. (ab - c)(ab + c) \\
&20. s(3 - 2s)(3 + 2s) \\
&21. (xy - 2z)(xy + 2z) \\
&22. 16(2t^2 - s^2)(2t^2 + s^2) \\
&23. (4x^2 - 6y)(4x^2 + 6y) \\
&24. 3(3a^2 - 2b)(3a^2 + 2b) \\
\\
&\text{Extension:} \\
&1. (x + 2)(3x - 2) \\
&2. 3(x + 5)(x - 1) \\
\end{aligned}
}
\]