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

Here are the solutions for 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. They are -2 and -6.
$(x - 2)(x - 6)$

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

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

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

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

6) $3a(a - 2) - 4a + 3$
* Simplify: Expand brackets and combine like terms ($-6a$ and $-4a$).
$3a^2 - 6a - 4a + 3 \rightarrow 3a^2 - 10a + 3$
* Factorise: Multiply first and last term ($3 \times 3 = 9$). Find factors of 9 that add to -10 (-9 and -1). Split the middle term.
$3a^2 - 9a - a + 3$
Factor pairs: $3a(a - 3) - 1(a - 3)$
$(3a - 1)(a - 3)$

7) $5w(w - 2) - 4w - 3$
* Simplify: Expand brackets and combine like terms ($-10w$ and $-4w$).
$5w^2 - 10w - 4w - 3 \rightarrow 5w^2 - 14w - 3$
* Factorise: Multiply first and last term ($5 \times -3 = -15$). Factors of -15 adding to -14 are -15 and +1.
$5w^2 - 15w + w - 3$
Factor pairs: $5w(w - 3) + 1(w - 3)$
$(5w + 1)(w - 3)$

8) $3(6 - 5s) + s^2 + s^2$
* Simplify: Expand brackets and combine $s^2$ terms.
$18 - 15s + 2s^2$
Rearrange: $2s^2 - 15s + 18$
* Factorise: Multiply first and last ($2 \times 18 = 36$). Factors of 36 adding to -15 are -12 and -3.
$2s^2 - 12s - 3s + 18$
Factor pairs: $2s(s - 6) - 3(s - 6)$
$(2s - 3)(s - 6)$

9) $3 + 2y(4y + 5)$
* Simplify: Expand brackets.
$3 + 8y^2 + 10y$
Rearrange: $8y^2 + 10y + 3$
* Factorise: Multiply first and last ($8 \times 3 = 24$). Factors of 24 adding to 10 are 6 and 4.
$8y^2 + 6y + 4y + 3$
Factor pairs: $2y(4y + 3) + 1(4y + 3)$
$(2y + 1)(4y + 3)$

10) $9x^2 - (x - 3)^2$
* Simplify: Expand the squared bracket carefully: $(x-3)(x-3) = x^2 - 6x + 9$.
$9x^2 - (x^2 - 6x + 9)$
Subtract everything inside: $9x^2 - x^2 + 6x - 9$
Combine: $8x^2 + 6x - 9$
* Factorise: Multiply first and last ($8 \times -9 = -72$). Factors of -72 adding to +6 are +12 and -6.
$8x^2 + 12x - 6x - 9$
Factor pairs: $4x(2x + 3) - 3(2x + 3)$
$(4x - 3)(2x + 3)$

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Section B: Factorise (Difference of Two Squares)

*Rule: $a^2 - b^2 = (a + b)(a - b)$*

1) $x^2 - 4 = (x + 2)(x - 2)$
2) $s^2 - 25 = (s + 5)(s - 5)$
3) $t^2 - 64 = (t + 8)(t - 8)$
4) $9 - y^2 = (3 + y)(3 - y)$
5) $49 - p^2 = (7 + p)(7 - p)$
6) $4q^2 - 121 = (2q + 11)(2q - 11)$
7) $81 - 25k^2 = (9 + 5k)(9 - 5k)$
8) $1 - 400d^2 = (1 + 20d)(1 - 20d)$
9) $600v^2 - 6$: First take out common factor 6 $\rightarrow 6(100v^2 - 1) = 6(10v + 1)(10v - 1)$
10) $a^2 - b^2 = (a + b)(a - b)$
11) $x^2 - 9y^2 = (x + 3y)(x - 3y)$
12) $4c^2 - d^2 = (2c + d)(2c - d)$
13) $16s^2 - 9t^2 = (4s + 3t)(4s - 3t)$
14) $49w^2 - 100v^2 = (7w + 10v)(7w - 10v)$
15) $32p^2 - 18q^2$: Take out common factor 2 $\rightarrow 2(16p^2 - 9q^2) = 2(4p + 3q)(4p - 3q)$
16) $48x^2 - 12y^2$: Take out common factor 12 $\rightarrow 12(4x^2 - y^2) = 12(2x + y)(2x - y)$
17) $45a^2 - 125b^2$: Take out common factor 5 $\rightarrow 5(9a^2 - 25b^2) = 5(3a + 5b)(3a - 5b)$
18) $72x^2 - 242y^2$: Take out common factor 2 $\rightarrow 2(36x^2 - 121y^2) = 2(6x + 11y)(6x - 11y)$
19) $a^2b^2 - c^2 = (ab + c)(ab - c)$
20) $9s - 4s^3$: Take out common factor $s \rightarrow s(9 - 4s^2) = s(3 + 2s)(3 - 2s)$
21) $(xy)^2 - 4z^2 = (xy + 2z)(xy - 2z)$
22) $64t^4 - 16s^4$: Take out common factor 16 $\rightarrow 16(4t^4 - s^4) = 16((2t^2)^2 - (s^2)^2) = 16(2t^2 + s^2)(2t^2 - s^2)$
23) $(4x^2)^2 - 36y^2$: This is $16x^4 - 36y^2$. Take out 4 $\rightarrow 4(4x^4 - 9y^2) = 4((2x^2)^2 - (3y)^2) = 4(2x^2 + 3y)(2x^2 - 3y)$
24) $27a^4 - 12b^2$: Take out common factor 3 $\rightarrow 3(9a^4 - 4b^2) = 3((3a^2)^2 - (2b)^2) = 3(3a^2 + 2b)(3a^2 - 2b)$

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Extension: Difference of Two Squares

1) $4x^2 - (x - 2)^2$
This is in the form $A^2 - B^2$ where $A = 2x$ and $B = (x - 2)$.
Formula: $(A + B)(A - B)$
Step 1: Add them $\rightarrow (2x + (x - 2)) = (3x - 2)$
Step 2: Subtract them $\rightarrow (2x - (x - 2)) = (2x - x + 2) = (x + 2)$
Answer: $(3x - 2)(x + 2)$

2) $(2x + 1)^2 - (x - 4)^2$
This is $A^2 - B^2$ where $A = (2x + 1)$ and $B = (x - 4)$.
Step 1: Add them $\rightarrow (2x + 1 + x - 4) = (3x - 3)$
Step 2: Subtract them $\rightarrow (2x + 1 - (x - 4)) = (2x + 1 - x + 4) = (x + 5)$
Combine: $(3x - 3)(x + 5)$
Note: You can factor out a 3 from the first bracket for a cleaner answer: $3(x - 1)(x + 5)$

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) $(2y + 1)(4y + 3)$
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) $2(6x + 11y)(6x - 11y)$
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) $4(2x^2 + 3y)(2x^2 - 3y)$
24) $3(3a^2 + 2b)(3a^2 - 2b)$

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