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Factorisation using identities · 3 marks

Let us try to factor 50p2 + 60pq + 18q2. What will a and b be in this case?

Answer: Take out the common factor 2 first: 50p2 + 60pq + 18q2 = 2(25p2 + 30pq + 9q2) = 2(5p + 3q)2, with a = 5p and b = 3q.

Step-by-step solution

Given: 50p2 + 60pq + 18q2
To find: Its factors, and the values of a and b in a2 + 2ab + b2

Idea: 50 and 18 are not perfect squares, so the identity does not fit directly (we would need √50 p). But 50, 60 and 18 are all even. Taking out the common factor 2 leaves 25, 30, 9, and 25 and 9 are perfect squares.

  1. 50p2 is not a perfect square of a nice term (its square root is √50 p). Look for a common factor instead: 2 divides 50, 60 and 18.½ mark
  2. 50p2 + 60pq + 18q2 = 2(25p2 + 30pq + 9q2)½ mark
  3. Inside the bracket: 25p2 = (5p)2 and 9q2 = (3q)2. So a = 5p, b = 3q.½ mark
  4. Middle term check: 2ab = 2(5p)(3q) = 30pq ✓.½ mark
  5. 50p2 + 60pq + 18q2 = 2[(5p)2 + 2(5p)(3q) + (3q)2]½ mark
  6. = 2(5p + 3q)2½ mark
50p² + 60pq + 18q² = 2(5p + 3q)², where a = 5p and b = 3q (after taking 2 out as a common factor).

Check: Put p = 1, q = 1: 50 + 60 + 18 = 128 and 2(5 + 3)2 = 2 × 64 = 128 ✓.

Answer to write in the exam

50p2 + 60pq + 18q2 = 2(25p2 + 30pq + 9q2)

= 2[(5p)2 + 2(5p)(3q) + (3q)2] [a2 + 2ab + b2 = (a + b)2]

∴ 50p2 + 60pq + 18q2 = 2(5p + 3q)2, with a = 5p, b = 3q

Common mistakes that cost marks

  • Forgetting to write the 2 in front of the final answer. (5p + 3q)2 alone is only 25p2 + 30pq + 9q2.
  • Writing 2(5p + 3q)2 as (10p + 6q)2. That equals 4(5p + 3q)2, double the correct value.
  • Giving up because 50 is not a perfect square, instead of checking for a common factor first.

How this can come in the exam

MCQ (1 mark)

The factorised form of 12x2 + 12x + 3 is

  1. (2x + 1)2
  2. 3(2x + 1)2
  3. 3(4x + 1)2
  4. (6x + 3)2
Show answer

(B) 3(2x + 1)2
12x2 + 12x + 3 = 3(4x2 + 4x + 1) = 3(2x + 1)2.

Short answer (2 marks)

Factorise 8m2 + 24mn + 18n2.

Show answerCommon factor 2: 2(4m2 + 12mn + 9n2) (1 mark). 4m2 = (2m)2, 9n2 = (3n)2, 12mn = 2(2m)(3n), so the answer is 2(2m + 3n)2 (1 mark).

Try one yourself

Factorise 75x2 + 60xy + 12y2.

Show answer

Common factor 3: 3(25x2 + 20xy + 4y2) = 3(5x + 2y)2.

More questions like this

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← Let us try to find factors of another algebraic expression: 36x² + 12xWhat if we replace b by −b in (a + b)² = a² + 2ab + b²? →