Factorise the difference between two squares with this Corbett Maths worksheet.

Let’s solve each part step by step.
We’re using the difference of two squares rule:
> $ a^2 - b^2 = (a - b)(a + b) $

This means if you have something squared minus another thing squared, you can factor it into two brackets: one with a minus and one with a plus.

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(a) $ x^2 - 25 $
→ $ 25 = 5^2 $, so this is $ x^2 - 5^2 $
→ Factor: $ (x - 5)(x + 5) $

(b) $ y^2 - 49 $
→ $ 49 = 7^2 $ → $ y^2 - 7^2 $
→ Factor: $ (y - 7)(y + 7) $

(c) $ w^2 - 100 $
→ $ 100 = 10^2 $ → $ w^2 - 10^2 $
→ Factor: $ (w - 10)(w + 10) $

(d) $ x^2 - 4 $
→ $ 4 = 2^2 $ → $ x^2 - 2^2 $
→ Factor: $ (x - 2)(x + 2) $

(e) $ c^2 - 64 $
→ $ 64 = 8^2 $ → $ c^2 - 8^2 $
→ Factor: $ (c - 8)(c + 8) $

(f) $ x^2 - 1 $
→ $ 1 = 1^2 $ → $ x^2 - 1^2 $
→ Factor: $ (x - 1)(x + 1) $

(g) $ x^2 - 900 $
→ $ 900 = 30^2 $ → $ x^2 - 30^2 $
→ Factor: $ (x - 30)(x + 30) $

(h) $ y^2 - 9 $
→ $ 9 = 3^2 $ → $ y^2 - 3^2 $
→ Factor: $ (y - 3)(y + 3) $

(i) $ 16 - x^2 $
→ This is same as $ -(x^2 - 16) $, but better to write as $ 4^2 - x^2 $
→ Factor: $ (4 - x)(4 + x) $ — or usually written as $ (4 - x)(4 + x) $, which is fine. Some prefer $ (4 + x)(4 - x) $ — order doesn’t matter.

(j) $ 1 - y^2 $
→ $ 1^2 - y^2 $ → Factor: $ (1 - y)(1 + y) $

(k) $ 81 - x^2 $
→ $ 9^2 - x^2 $ → Factor: $ (9 - x)(9 + x) $

(l) $ 144 - h^2 $
→ $ 12^2 - h^2 $ → Factor: $ (12 - h)(12 + h) $

(m) $ x^2 - y^2 $
→ Already in form $ a^2 - b^2 $ → Factor: $ (x - y)(x + y) $

(n) $ a^2 - c^2 $
→ Same idea → Factor: $ (a - c)(a + c) $

(o) $ 9x^2 - 25 $
→ $ 9x^2 = (3x)^2 $, $ 25 = 5^2 $ → So $ (3x)^2 - 5^2 $
→ Factor: $ (3x - 5)(3x + 5) $

(p) $ 4y^2 - 1 $
→ $ 4y^2 = (2y)^2 $, $ 1 = 1^2 $ → $ (2y)^2 - 1^2 $
→ Factor: $ (2y - 1)(2y + 1) $

(q) $ 49x^2 - 16 $
→ $ 49x^2 = (7x)^2 $, $ 16 = 4^2 $ → $ (7x)^2 - 4^2 $
→ Factor: $ (7x - 4)(7x + 4) $

(r) $ 100 - 81x^2 $
→ $ 100 = 10^2 $, $ 81x^2 = (9x)^2 $ → $ 10^2 - (9x)^2 $
→ Factor: $ (10 - 9x)(10 + 9x) $

(s) $ 9x^2 - 4y^2 $
→ $ 9x^2 = (3x)^2 $, $ 4y^2 = (2y)^2 $ → $ (3x)^2 - (2y)^2 $
→ Factor: $ (3x - 2y)(3x + 2y) $

(t) $ 36a^2 - c^2 $
→ $ 36a^2 = (6a)^2 $, $ c^2 = c^2 $ → $ (6a)^2 - c^2 $
→ Factor: $ (6a - c)(6a + c) $

(u) $ 121w^2 - 196y^2 $
→ $ 121w^2 = (11w)^2 $, $ 196y^2 = (14y)^2 $ → $ (11w)^2 - (14y)^2 $
→ Factor: $ (11w - 14y)(11w + 14y) $

(v) $ 225 - 121y^2 $
→ $ 225 = 15^2 $, $ 121y^2 = (11y)^2 $ → $ 15^2 - (11y)^2 $
→ Factor: $ (15 - 11y)(15 + 11y) $

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Final Answer:
(a) (x - 5)(x + 5)
(b) (y - 7)(y + 7)
(c) (w - 10)(w + 10)
(d) (x - 2)(x + 2)
(e) (c - 8)(c + 8)
(f) (x - 1)(x + 1)
(g) (x - 30)(x + 30)
(h) (y - 3)(y + 3)
(i) (4 - x)(4 + x)
(j) (1 - y)(1 + y)
(k) (9 - x)(9 + x)
(l) (12 - h)(12 + h)
(m) (x - y)(x + y)
(n) (a - c)(a + c)
(o) (3x - 5)(3x + 5)
(p) (2y - 1)(2y + 1)
(q) (7x - 4)(7x + 4)
(r) (10 - 9x)(10 + 9x)
(s) (3x - 2y)(3x + 2y)
(t) (6a - c)(6a + c)
(u) (11w - 14y)(11w + 14y)
(v) (15 - 11y)(15 + 11y)