Comprehensive algebra worksheet covering the simplification and factorization of various quadratic expressions.

Let’s solve each problem step by step. We’ll start with Section A, then Section B, and finally the Extension.

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Section A: Simplify then factorise

We need to first simplify each expression (expand brackets, combine like terms), then factorise the resulting quadratic.

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1) x² – 6x – 2x + 12

Simplify:
x² – 8x + 12

Factorise: Find two numbers that multiply to 12 and add to -8 → -6 and -2
→ (x – 6)(x – 2)

✔ Final: (x – 6)(x – 2)

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2) d(d – 5) – 84

Expand:
d² – 5d – 84

Factorise: Two numbers that multiply to -84 and add to -5 → -12 and +7
→ (d – 12)(d + 7)

✔ Final: (d – 12)(d + 7)

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3) b² + 2(b – 4)

Expand:
b² + 2b – 8

Factorise: Multiply to -8, add to +2 → +4 and -2
→ (b + 4)(b – 2)

✔ Final: (b + 4)(b – 2)

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4) x² – 3(2x + 9)

Expand:
x² – 6x – 27

Factorise: Multiply to -27, add to -6 → -9 and +3
→ (x – 9)(x + 3)

✔ Final: (x – 9)(x + 3)

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5) c(c + 8) – 48

Expand:
c² + 8c – 48

Factorise: Multiply to -48, add to +8 → +12 and -4
→ (c + 12)(c – 4)

✔ Final: (c + 12)(c – 4)

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6) 3a(a – 2) – 4a + 3

Expand:
3a² – 6a – 4a + 3 = 3a² – 10a + 3

Factorise: Multiply 3×3=9, find factors of 9 that add to -10 → -9 and -1
Split middle term:
3a² – 9a – a + 3
Group: (3a² – 9a) + (-a + 3) = 3a(a – 3) –1(a – 3)
→ (3a – 1)(a – 3)

✔ Final: (3a – 1)(a – 3)

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7) 5w(w – 2) – 4w – 3

Expand:
5w² – 10w – 4w – 3 = 5w² – 14w – 3

Factorise: Multiply 5×(-3)= -15, find factors of -15 that add to -14 → -15 and +1
Split: 5w² – 15w + w – 3
Group: 5w(w – 3) +1(w – 3)
→ (5w + 1)(w – 3)

✔ Final: (5w + 1)(w – 3)

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8) 3(6 – 5s) + s² + s²

Simplify:
18 – 15s + 2s² → rearrange: 2s² – 15s + 18

Factorise: Multiply 2×18=36, factors of 36 that add to -15 → -12 and -3
Split: 2s² – 12s – 3s + 18
Group: 2s(s – 6) –3(s – 6)
→ (2s – 3)(s – 6)

✔ Final: (2s – 3)(s – 6)

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9) 3 + 2y(4y + 5)

Expand:
3 + 8y² + 10y → rearrange: 8y² + 10y + 3

Factorise: Multiply 8×3=24, factors of 24 that add to 10 → 6 and 4
Split: 8y² + 6y + 4y + 3
Group: 2y(4y + 3) +1(4y + 3)
→ (2y + 1)(4y + 3)

✔ Final: (2y + 1)(4y + 3)

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10) 9x² – (x – 3)²

First expand (x – 3)² = x² – 6x + 9
So: 9x² – (x² – 6x + 9) = 9x² – x² + 6x – 9 = 8x² + 6x – 9

Factorise: Multiply 8×(-9)= -72, factors of -72 that add to +6 → +12 and -6
Split: 8x² + 12x – 6x – 9
Group: 4x(2x + 3) –3(2x + 3)
→ (4x – 3)(2x + 3)

✔ Final: (4x – 3)(2x + 3)

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Section B: Factorise using difference of squares or other methods

Remember: Difference of squares: a² – b² = (a – b)(a + b)

Also look for common factors first!

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1) x² – 4 = (x – 2)(x + 2)

✔ Final: (x – 2)(x + 2)

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2) s² – 25 = (s – 5)(s + 5)

✔ Final: (s – 5)(s + 5)

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3) t² – 64 = (t – 8)(t + 8)

✔ Final: (t – 8)(t + 8)

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4) 9 – y² = (3 – y)(3 + y)

✔ Final: (3 – y)(3 + y)

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5) 49 – p² = (7 – p)(7 + p)

✔ Final: (7 – p)(7 + p)

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6) 4q² – 121 = (2q)² – 11² = (2q – 11)(2q + 11)

✔ Final: (2q – 11)(2q + 11)

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7) 81 – 25k² = 9² – (5k)² = (9 – 5k)(9 + 5k)

✔ Final: (9 – 5k)(9 + 5k)

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8) 1 – 400d² = 1² – (20d)² = (1 – 20d)(1 + 20d)

✔ Final: (1 – 20d)(1 + 20d)

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9) 600v² – 6

Factor out GCF: 6(100v² – 1) = 6[(10v)² – 1²] = 6(10v – 1)(10v + 1)

✔ Final: 6(10v – 1)(10v + 1)

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10) a² – b² = (a – b)(a + b)

✔ Final: (a – b)(a + b)

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11) x² – 9y² = x² – (3y)² = (x – 3y)(x + 3y)

✔ Final: (x – 3y)(x + 3y)

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12) 4c² – d² = (2c)² – d² = (2c – d)(2c + d)

✔ Final: (2c – d)(2c + d)

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13) 16s² – 9t² = (4s)² – (3t)² = (4s – 3t)(4s + 3t)

✔ Final: (4s – 3t)(4s + 3t)

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14) 49w² – 100v² = (7w)² – (10v)² = (7w – 10v)(7w + 10v)

✔ Final: (7w – 10v)(7w + 10v)

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15) 32p² – 18q²

GCF is 2: 2(16p² – 9q²) = 2[(4p)² – (3q)²] = 2(4p – 3q)(4p + 3q)

✔ Final: 2(4p – 3q)(4p + 3q)

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16) 48x² – 12y²

GCF is 12: 12(4x² – y²) = 12[(2x)² – y²] = 12(2x – y)(2x + y)

✔ Final: 12(2x – y)(2x + y)

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17) 45a² – 125b²

GCF is 5: 5(9a² – 25b²) = 5[(3a)² – (5b)²] = 5(3a – 5b)(3a + 5b)

✔ Final: 5(3a – 5b)(3a + 5b)

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18) 72x² – 242y²

GCF is 2: 2(36x² – 121y²) = 2[(6x)² – (11y)²] = 2(6x – 11y)(6x + 11y)

✔ Final: 2(6x – 11y)(6x + 11y)

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19) a²b² – c² = (ab)² – c² = (ab – c)(ab + c)

✔ Final: (ab – c)(ab + c)

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20) 9s – 4s³

Factor out s: s(9 – 4s²) = s[3² – (2s)²] = s(3 – 2s)(3 + 2s)

✔ Final: s(3 – 2s)(3 + 2s)

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21) (xy)² – 4z² = (xy)² – (2z)² = (xy – 2z)(xy + 2z)

✔ Final: (xy – 2z)(xy + 2z)

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22) 64t⁴ – 16s⁴

GCF is 16: 16(4t⁴ – s⁴) → now 4t⁴ – s = (2t²)² – (s²)² = (2t² – s²)(2t² + s²)

But wait — can we go further? 2t² – s² is not a difference of squares unless coefficients are perfect squares. So stop here.

Actually, let’s check original: 64t⁴ – 16s⁴ = 16(4t⁴ – s⁴) = 16[(2t²)² – (s²)²] = 16(2t² – s²)(2t² + s²)

Yes, that’s fully factored over integers.

✔ Final: 16(2t² – s²)(2t² + s²)

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23) (4x²)² – 36y² = 16x⁴ – 36y²

GCF is 4: 4(4x⁴ – 9y²) = 4[(2x²)² – (3y)²] = 4(2x² – 3y)(2x² + 3y)

✔ Final: 4(2x² – 3y)(2x² + 3y)

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24) 27a⁴ – 12b²

GCF is 3: 3(9a⁴ – 4b²) = 3[(3a²)² – (2b)²] = 3(3a² – 2b)(3a² + 2b)

✔ Final: 3(3a² – 2b)(3a² + 2b)

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Extension: Using difference of two squares

These are already in form A² – B², so factor directly.

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1) 4x² – (x – 2)²

This is [2x]² – [x – 2]² → difference of squares!

= (2x – (x – 2))(2x + (x – 2))
= (2x – x + 2)(2x + x – 2)
= (x + 2)(3x – 2)

✔ Final: (x + 2)(3x – 2)

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2) (2x + 1)² – (x – 4)²

Difference of squares: A² – B² = (A – B)(A + B)

A = 2x + 1, B = x – 4

A – B = (2x + 1) – (x – 4) = 2x + 1 – x + 4 = x + 5
A + B = (2x + 1) + (x – 4) = 3x – 3

So: (x + 5)(3x – 3)

Can factor 3 from second bracket: 3(x + 5)(x – 1)

✔ Final: 3(x + 5)(x – 1)

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Final Answer:

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 + 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(2t² – s²)(2t² + s²)
23) 4(2x² – 3y)(2x² + 3y)
24) 3(3a² – 2b)(3a² + 2b)

Extension:
1) (x + 2)(3x – 2)
2) 3(x + 5)(x – 1)