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Algebraic identities · 2 marks

What if we replace b by −b in (a + b)2 = a2 + 2ab + b2?

Answer: We get (a − b)2 = a2 − 2ab + b2, which is also an identity.

Step-by-step solution

Given: (a + b)2 = a2 + 2ab + b2, true for all numbers a, b
To find: The result when b is replaced by −b

Idea: An identity is true for every value of b, so it is also true when b is replaced by −b. Replace b by −b everywhere, then simplify the signs.

  1. Write −b in place of every b:
    (a + (−b))2 = a2 + 2a(−b) + (−b)2½ mark
  2. Left side: a + (−b) = a − b.½ mark
  3. Right side: 2a(−b) = −2ab, and (−b)2 = b2 (a negative times a negative is positive).½ mark
  4. So (a − b)2 = a2 − 2ab + b2. It is true for all a and b, so it is also an identity.½ mark
Replacing b by −b gives (a − b)² = a² − 2ab + b², a new identity.

Check: a = 10, b = 3: (10 − 3)2 = 49 and 100 − 60 + 9 = 49 ✓. Also by the distributive property: (a − b)(a − b) = a2 − ab − ba + b2 = a2 − 2ab + b2 ✓.

Answer to write in the exam

[a + (−b)]2 = a2 + 2a(−b) + (−b)2 [(a + b)2 = a2 + 2ab + b2]

∴ (a − b)2 = a2 − 2ab + b2

Common mistakes that cost marks

  • Writing (−b)2 = −b2. The square of −b is +b2.
  • Writing (a − b)2 = a2 − b2. That is a different expression; (a − b)2 has three terms.
  • Writing (a − b)2 = a2 − 2ab − b2. Only the middle term changes sign; (−b)2 = +b2.

How this can come in the exam

MCQ (1 mark)

(x − 5)2 is equal to

  1. x2 − 25
  2. x2 − 10x − 25
  3. x2 − 10x + 25
  4. x2 + 10x + 25
Show answer

(C) x2 − 10x + 25
(a − b)2 = a2 − 2ab + b2 with a = x, b = 5.

Assertion–Reason (1 mark)

Assertion (A): (a − b)2 = a2 − 2ab + b2 for all a, b.
Reason (R): An identity stays true when a variable is replaced by its negative.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
Replacing b by −b in (a + b)2 = a2 + 2ab + b2 gives A directly, so R explains A.

Try one yourself

Use (a − b)2 = a2 − 2ab + b2 to expand (3x − 4y)2.

Show answer

(3x)2 − 2(3x)(4y) + (4y)2 = 9x2 − 24xy + 16y2.

More questions like this

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