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

It looks like you've uploaded a worksheet titled "Factorising Expressions" from Corbettmaths, which contains two sections (A and B) of algebraic expressions to factorize fully. Since I can't view images directly, I’ll help you solve the problems based on the text you’ve provided in your message.

Let’s go through each question step by step, explaining how to factorize each expression fully.

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🔷 Section A

#### 1) $ 8x + 24 $
- Both terms have a common factor: 8
- $ = 8(x + 3) $

#### 2) $ 15 + 25y $
- Common factor: 5
- $ = 5(3 + 5y) $

#### 3) $ 32 - 40w $
- Common factor: 8
- $ = 8(4 - 5w) $

#### 4) $ 16a - 36 $
- Common factor: 4
- $ = 4(4a - 9) $

#### 5) $ 16d^2 - 4d $
- Common factor: 4d
- $ = 4d(4d - 1) $

#### 6) $ 12a + 60a^2 $
- Common factor: 6a
- $ = 6a(2 + 10a) $ → Wait! Let's check:
$ 6a \times 2 = 12a $, $ 6a \times 10a = 60a^2 $ → correct
But we can factor out more: actually, GCF is 6a, but let's write it properly:
$ = 6a(2 + 10a) $ — but 10a is not simplified. Actually, we should factor out 6a and get:
$ = 6a(2 + 10a) $ → no, that’s not fully factored? Wait:
Actually, $ 12a + 60a^2 = 6a(2 + 10a) $? No, better:
$ = 6a(2 + 10a) $ → but 2 and 10 have a common factor, so we can do better?
No — the GCF is 6a, so:
$ = 6a(2 + 10a) $ → but this is not simplified. Wait, actually:
$ 12a + 60a^2 = 12a(1 + 5a) $? Let's check:
$ 12a \times 1 = 12a $, $ 12a \times 5a = 60a^2 $ → YES!
So correct factorization: $ = 12a(1 + 5a) $

✔ Correct: $ 12a + 60a^2 = 12a(1 + 5a) $

#### 7) $ 21xy + 14x $
- Common factor: 7x
- $ = 7x(3y + 2) $

#### 8) $ 27ab - 18a^2 $
- Common factor: 9a
- $ = 9a(3b - 2a) $

#### 9) $ 12a^4 + 28a $
- Common factor: 4a
- $ = 4a(3a^3 + 7) $

#### 10) $ 72wx + 45w^2x $
- Common factor: 9wx
- $ = 9wx(8 + 5w) $

#### 11) $ 22x^2y - 55xy^2 $
- Common factor: 11xy
- $ = 11xy(2x - 5y) $

#### 12) $ 16h^4 + 24h^3 $
- Common factor: 8h^3
- $ = 8h^3(2h + 3) $

#### 13) $ 9h^2g - 15h^3 $
- Common factor: 3h^2
- $ = 3h^2(3g - 5h) $

#### 14) $ 12c^3d^2 + 20c^2d $
- Common factor: 4c^2d
- $ = 4c^2d(3cd + 5) $

#### 15) $ 28a^3b^2 - 7a^2b $
- Common factor: 7a^2b
- $ = 7a^2b(4ab - 1) $

#### 16) $ 60a^2b^3 - 35a^2b^2 $
- Common factor: 5a^2b^2
- $ = 5a^2b^2(12b - 7) $

#### 17) $ 88a^3b + 56a^2b^2 $
- Common factor: 8a^2b
- $ = 8a^2b(11a + 7b) $

#### 18) $ 36p^2q^3 - 48p^3q^2 $
- Common factor: 12p^2q^2
- $ = 12p^2q^2(3q - 4p) $

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🔷 Section B

Now for more complex ones — some may involve difference of squares, trinomial factoring, or grouping.

#### 1) $ 6 - 12gh + 3h $
- Rearranging: $ 6 + 3h - 12gh $
- Group terms: $ (6 + 3h) - 12gh $
- Better: look for common factors
- All terms divisible by 3:
$ = 3(2 + h - 4gh) $
- Can we factor further? Not easily.
Try grouping: $ 3(2 + h - 4gh) $ — no common binomial
So final answer: $ = 3(2 + h - 4gh) $

But wait — maybe rearrange: $ 6 + 3h - 12gh $
Try factoring by grouping:
Group as: $ (6 - 12gh) + (3h) $ — not helpful
Or: $ 3(2 + h - 4gh) $ — seems simplest.

✔ $ = 3(2 + h - 4gh) $

#### 2) $ 21st - 7t + 14 $
- Factor out 7: $ = 7(3st - t + 2) $
- Now look inside: $ 3st - t + 2 $
- Group: $ (3st - t) + 2 = t(3s - 1) + 2 $ — no common factor
- So cannot factor further
✔ $ = 7(3st - t + 2) $

#### 3) $ 32 - 44aw + 11v $
- Check for common factors: all coefficients divisible by... 32, 44, 11 — GCF is 1
- Terms: $ 32, -44aw, 11v $
- No common factor across all three
- Cannot factor further
✔ $ = 32 - 44aw + 11v $ → already simplified

Wait — perhaps typo? Or maybe group differently?

But no clear grouping. So likely cannot be factored further.

✔ $ = 32 - 44aw + 11v $ — no factorization possible

#### 4) $ 4ab + 2b - abc $
- Group: $ (4ab + 2b) - abc $
- Factor first group: $ 2b(2a + 1) - abc $
- Still not helpful
- Look for common factors: only b appears in all terms
- $ = b(4a + 2 - ac) $
✔ $ = b(4a + 2 - ac) $

#### 5) $ 5auv - 10av + 15au $
- Common factor: 5a
- $ = 5a(uv - 2v + 3u) $
- Inside: $ uv - 2v + 3u $ — can we factor?
Try grouping: $ (uv - 2v) + 3u = v(u - 2) + 3u $ — no common binomial
- So stop here
✔ $ = 5a(uv - 2v + 3u) $

#### 6) $ 16ay + 24y - 8xyz $
- Common factor: 8y
- $ = 8y(2a + 3 - xz) $
✔ $ = 8y(2a + 3 - xz) $

#### 7) $ 9nu - 27nau + 45w $
- Common factor: 9n? Only first two terms have n
- First two: $ 9nu - 27nau = 9nu(1 - 3a) $
- Third term: $ 45w $ — no common factor with others
- So overall: $ 9nu(1 - 3a) + 45w $
- No common factor across all
- So no full factorization possible
✔ $ = 9nu(1 - 3a) + 45w $ — or leave as is

But maybe factor 9? Yes: $ = 9(nu - 3nau + 5w) $
- That’s better
✔ $ = 9(nu - 3nau + 5w) $

#### 8) $ 24gh - 12g + 15h $
- Try factoring by grouping:
Group: $ (24gh - 12g) + 15h = 12g(2h - 1) + 15h $
- No common factor
- GCF of all terms: 3
- $ = 3(8gh - 4g + 5h) $
- Inside: $ 8gh - 4g + 5h $ — try grouping:
$ (8gh - 4g) + 5h = 4g(2h - 1) + 5h $ — still no
- So final: $ = 3(8gh - 4g + 5h) $

✔ $ = 3(8gh - 4g + 5h) $

#### 9) $ 132pqr - 96qr + 108pqrs $
- Look at coefficients: 132, 96, 108 — find GCF
- GCF of 132, 96, 108:
- 132 = 12×11, 96=12×8, 108=12×9 → GCF = 12
- Variables: all have q, r, but not all have p or s
- Common variables: q and r — both terms have qr
- So common factor: 12qr
- $ = 12qr(11p - 8 + 9ps) $
- Because:
- $ 12qr × 11p = 132pqr $
- $ 12qr × (-8) = -96qr $
- $ 12qr × 9ps = 108pqrs $
✔ $ = 12qr(11p - 8 + 9ps) $

#### 10) $ 2x + xy - x^2 $
- Rearrange: $ -x^2 + xy + 2x $
- Factor out x: $ = x(-x + y + 2) $
- Or write: $ = x(-x + y + 2) $
- Or: $ = x(y + 2 - x) $
✔ $ = x(y + 2 - x) $

#### 11) $ 5k^2 - 10jk + k $
- Factor out k: $ = k(5k - 10j + 1) $
- Inside: $ 5k - 10j + 1 $ — no common factor
✔ $ = k(5k - 10j + 1) $

#### 12) $ 9cd - 3c^2d + 12c $
- Factor out 3c: $ = 3c(3d - cd + 4) $
- Inside: $ 3d - cd + 4 $
- Group: $ (3d - cd) + 4 = d(3 - c) + 4 $ — no
- So stop here
✔ $ = 3c(3d - cd + 4) $

#### 13) $ 7xyz + xy^2 - x^2y $
- Common factor: xy
- $ = xy(7z + y - x) $
✔ $ = xy(7z + y - x) $

#### 14) $ w^2 - 5w^2t^2 + w^2 $
- Combine like terms: $ w^2 + w^2 = 2w^2 $
- So: $ 2w^2 - 5w^2t^2 $
- Factor: $ = w^2(2 - 5t^2) $
✔ $ = w^2(2 - 5t^2) $

#### 15) $ 8st^2 - 32s^2t + 64st $
- Common factor: 8st
- $ = 8st(t - 4s + 8) $
- Check:
- $ 8st × t = 8st^2 $
- $ 8st × (-4s) = -32s^2t $
- $ 8st × 8 = 64st $
✔ $ = 8st(t - 4s + 8) $

#### 16) $ 12gh^2 - 9gh^3 + 18gh $
- Common factor: 3gh
- $ = 3gh(4h - 3h^2 + 6) $
- Rearranged: $ = 3gh(-3h^2 + 4h + 6) $
- Or write: $ = 3gh(4h - 3h^2 + 6) $
- But better order: $ = 3gh(-3h^2 + 4h + 6) $
- Or factor out negative: $ = -3gh(3h^2 - 4h - 6) $
- But unless quadratic factors nicely, leave as:
✔ $ = 3gh(4h - 3h^2 + 6) $ — or $ = 3gh(-3h^2 + 4h + 6) $

#### 17) $ jab + \frac{1}{2}a - a $
- Simplify: $ jab + (\frac{1}{2}a - a) = jab - \frac{1}{2}a $
- Factor: $ = a(jb - \frac{1}{2}) $
- Or write: $ = a\left(jb - \frac{1}{2}\right) $
✔ $ = a\left(jb - \frac{1}{2}\right) $

#### 18) $ \frac{1}{2}xy + x^2y^2 + \frac{1}{2}x^3y $
- Common factor: $ \frac{1}{2}xy $
- $ = \frac{1}{2}xy(1 + 2xy + x^2) $
- Check:
- $ \frac{1}{2}xy × 1 = \frac{1}{2}xy $
- $ \frac{1}{2}xy × 2xy = x^2y^2 $
- $ \frac{1}{2}xy × x^2 = \frac{1}{2}x^3y $
✔ $ = \frac{1}{2}xy(1 + 2xy + x^2) $

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✔ Final Answers Summary:

#### Section A:
1) $ 8(x + 3) $
2) $ 5(3 + 5y) $
3) $ 8(4 - 5w) $
4) $ 4(4a - 9) $
5) $ 4d(4d - 1) $
6) $ 12a(1 + 5a) $
7) $ 7x(3y + 2) $
8) $ 9a(3b - 2a) $
9) $ 4a(3a^3 + 7) $
10) $ 9wx(8 + 5w) $
11) $ 11xy(2x - 5y) $
12) $ 8h^3(2h + 3) $
13) $ 3h^2(3g - 5h) $
14) $ 4c^2d(3cd + 5) $
15) $ 7a^2b(4ab - 1) $
16) $ 5a^2b^2(12b - 7) $
17) $ 8a^2b(11a + 7b) $
18) $ 12p^2q^2(3q - 4p) $

#### Section B:
1) $ 3(2 + h - 4gh) $
2) $ 7(3st - t + 2) $
3) $ 32 - 44aw + 11v $ (no further factorization)
4) $ b(4a + 2 - ac) $
5) $ 5a(uv - 2v + 3u) $
6) $ 8y(2a + 3 - xz) $
7) $ 9(nu - 3nau + 5w) $
8) $ 3(8gh - 4g + 5h) $
9) $ 12qr(11p - 8 + 9ps) $
10) $ x(y + 2 - x) $
11) $ k(5k - 10j + 1) $
12) $ 3c(3d - cd + 4) $
13) $ xy(7z + y - x) $
14) $ w^2(2 - 5t^2) $
15) $ 8st(t - 4s + 8) $
16) $ 3gh(-3h^2 + 4h + 6) $
17) $ a\left(jb - \frac{1}{2}\right) $
18) $ \frac{1}{2}xy(1 + 2xy + x^2) $

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