Comprehensive worksheet designed to help students master factoring quadratic expressions through varied practice problems.

Here are the solutions to the problems on the worksheet.

Section A: Simplify then factorise

1) $x^2 - 6x - 2x + 12$
* Simplify: Combine like terms ($-6x$ and $-2x$).
$x^2 - 8x + 12$
* Factorise: Find two numbers that multiply to $12$ and add to $-8$. Those numbers are $-2$ and $-6$.
$(x - 2)(x - 6)$

2) $d(d - 5) - 84$
* Simplify: Expand the bracket.
$d^2 - 5d - 84$
* Factorise: Find two numbers that multiply to $-84$ and add to $-5$. Those numbers are $-12$ and $7$.
$(d - 12)(d + 7)$

3) $b^2 + 2(b - 4)$
* Simplify: Expand the bracket.
$b^2 + 2b - 8$
* Factorise: Find two numbers that multiply to $-8$ and add to $2$. Those numbers are $4$ and $-2$.
$(b + 4)(b - 2)$

4) $x^2 - 3(2x + 9)$
* Simplify: Expand the bracket (watch the negative sign).
$x^2 - 6x - 27$
* Factorise: Find two numbers that multiply to $-27$ and add to $-6$. Those numbers are $-9$ and $3$.
$(x - 9)(x + 3)$

5) $c(c + 8) - 48$
* Simplify: Expand the bracket.
$c^2 + 8c - 48$
* Factorise: Find two numbers that multiply to $-48$ and add to $8$. Those numbers are $12$ and $-4$.
$(c + 12)(c - 4)$

6) $3a(a - 2) - 4a + 3$
* Simplify: Expand the bracket and combine like terms ($-6a$ and $-4a$).
$3a^2 - 6a - 4a + 3 \rightarrow 3a^2 - 10a + 3$
* Factorise: We need factors of $3 \times 3 = 9$ that add to $-10$. They are $-9$ and $-1$.
Split the middle term: $3a^2 - 9a - a + 3$
Group: $3a(a - 3) - 1(a - 3)$
$(3a - 1)(a - 3)$

7) $5w(w - 2) - 4w - 3$
* Simplify: Expand and combine ($-10w$ and $-4w$).
$5w^2 - 10w - 4w - 3 \rightarrow 5w^2 - 14w - 3$
* Factorise: Factors of $5 \times -3 = -15$ that add to $-14$. They are $-15$ and $1$.
Split: $5w^2 - 15w + w - 3$
Group: $5w(w - 3) + 1(w - 3)$
$(5w + 1)(w - 3)$

8) $3(6 - 5s) + s^2 + s^2$
* Simplify: Expand and combine $s^2$ terms.
$18 - 15s + 2s^2 \rightarrow 2s^2 - 15s + 18$
* Factorise: Factors of $2 \times 18 = 36$ that add to $-15$. They are $-12$ and $-3$.
Split: $2s^2 - 12s - 3s + 18$
Group: $2s(s - 6) - 3(s - 6)$
$(2s - 3)(s - 6)$

9) $3 + 2y(4y + 5)$
* Simplify: Expand.
$3 + 8y^2 + 10y \rightarrow 8y^2 + 10y + 3$
* Factorise: Factors of $8 \times 3 = 24$ that add to $10$. They are $6$ and $4$.
Split: $8y^2 + 4y + 6y + 3$
Group: $4y(2y + 1) + 3(2y + 1)$
$(4y + 3)(2y + 1)$

10) $9x^2 - (x - 3)^2$
* Simplify: Expand the squared bracket first: $(x-3)(x-3) = x^2 - 6x + 9$.
$9x^2 - (x^2 - 6x + 9)$
Be careful with the minus sign outside the bracket:
$9x^2 - x^2 + 6x - 9$
$8x^2 + 6x - 9$
* Factorise: Factors of $8 \times -9 = -72$ that add to $6$. They are $12$ and $-6$.
Split: $8x^2 + 12x - 6x - 9$
Group: $4x(2x + 3) - 3(2x + 3)$
$(4x - 3)(2x + 3)$

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Section B: Factorise (Difference of Two Squares)

*Note: Most of these use the rule $a^2 - b^2 = (a - b)(a + b)$. Some require taking out a common number first.*

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) Take out 6 first: $6(100v^2 - 1) \rightarrow 6(10v - 1)(10v + 1)$
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) Take out 2 first: $2(16p^2 - 9q^2) \rightarrow 2(4p - 3q)(4p + 3q)$
16) Take out 12 first: $12(4x^2 - y^2) \rightarrow 12(2x - y)(2x + y)$

17) Take out 5 first: $5(9a^2 - 25b^2) \rightarrow 5(3a - 5b)(3a + 5b)$
18) Take out 18 first: $18(4x^2 - 13y^2)$. *Note: 13 is not a square number, so it stops here unless using surds.* Answer: $18(4x^2 - 13y^2)$
19) $(ab - c)(ab + c)$
20) Take out $s$ first: $s(9 - 4s^2) \rightarrow s(3 - 2s)(3 + 2s)$
21) $(xy - 2z)(xy + 2z)$
22) Take out $16s^4$? No, let's look closer. $64t^4 - 16s^4$. Take out 16: $16(4t^4 - s^4)$. Inside is difference of squares: $16((2t^2)^2 - (s^2)^2) \rightarrow 16(2t^2 - s^2)(2t^2 + s^2)$
23) Expand inside first or treat as squares. $(4x^2)^2 - (6y)^2$.
$(4x^2 - 6y)(4x^2 + 6y)$.
You can simplify further by taking out 2 from each bracket: $2(2x^2 - 3y) \cdot 2(2x^2 + 3y) \rightarrow 4(2x^2 - 3y)(2x^2 + 3y)$
24) Take out 3 first: $3(9a^4 - 4b^2)$. Inside is difference of squares: $(3a^2)^2 - (2b)^2$.
$3(3a^2 - 2b)(3a^2 + 2b)$

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Extension

1) $4x^2 - (x - 2)^2$
This is in the form $A^2 - B^2$ where $A = 2x$ and $B = (x - 2)$.
Formula: $(A - B)(A + B)$
$(2x - (x - 2)) (2x + (x - 2))$
Simplify the brackets:
$(2x - x + 2)(2x + x - 2)$
$(x + 2)(3x - 2)$

2) $(2x + 1)^2 - (x - 4)^2$
This is $A^2 - B^2$ where $A = (2x + 1)$ and $B = (x - 4)$.
Formula: $(A - B)(A + B)$
$((2x + 1) - (x - 4)) ((2x + 1) + (x - 4))$
Simplify the first bracket (watch the double negative): $(2x + 1 - x + 4) = (x + 5)$
Simplify the second bracket: $(2x + 1 + x - 4) = (3x - 3)$
Result: $(x + 5)(3x - 3)$
*Optional:* You can take out a 3 from the second bracket: $3(x + 5)(x - 1)$

Final Answer:
Section A
1) $(x - 2)(x - 6)$
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) $(4y + 3)(2y + 1)$
10) $(4x - 3)(2x + 3)$

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) $18(4x^2 - 13y^2)$
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) $4(2x^2 - 3y)(2x^2 + 3y)$
24) $3(3a^2 - 2b)(3a^2 + 2b)$

Extension
1) $(x + 2)(3x - 2)$
2) $3(x + 5)(x - 1)$