Learnify Academy is a tuition centre in Bahrain. Classes are for students in Bahrain only.Tuition classes in Bahrain only

Algebraic identities · 2 marks

By now you must have observed that (a + b)2 ≠ a2 + b2. But can you find out which one of them will be greater? For example, if a = 10 and b = 2, what are the values of (a + b)2 and a2 + b2?

Answer: (a + b)2 = 122 = 144 and a2 + b2 = 100 + 4 = 104, so here (a + b)2 is greater, by 2ab = 40. It is not greater for all numbers: it depends on the sign of 2ab.

Step-by-step solution

Given: a = 10, b = 2
To find: (a + b)2 and a2 + b2, and which is greater

Idea: Since (a + b)2 = a2 + 2ab + b2, the two expressions differ by exactly 2ab. Whichever is greater depends on whether 2ab is positive, negative or zero.

  1. (a + b)2 = (10 + 2)2 = 122 = 144½ mark
  2. a2 + b2 = 102 + 22 = 100 + 4 = 104½ mark
  3. 144 > 104, so in this case (a + b)2 > a2 + b2. The gap is 144 − 104 = 40 = 2 × 10 × 2 = 2ab.½ mark
  4. Is it true for all numbers? No. Take a = 10, b = −2: (a + b)2 = 82 = 64 but a2 + b2 = 104, so now a2 + b2 is greater. Here 2ab = −40 is negative.½ mark
For a = 10, b = 2: (a + b)² = 144 and a² + b² = 104, so (a + b)² is greater (by 2ab = 40). In general, (a + b)² is greater when ab > 0, smaller when ab < 0, and equal when ab = 0.

Check: Difference = (a + b)2 − (a2 + b2) = 2ab = 2 × 10 × 2 = 40, and 144 − 104 = 40 ✓.

Answer to write in the exam

(a + b)2 = (10 + 2)2 = 122 = 144

a2 + b2 = 102 + 22 = 100 + 4 = 104

144 − 104 = 40 = 2ab

In general, (a + b)2 − (a2 + b2) = 2ab: (a + b)2 is greater when ab > 0, smaller when ab < 0

∴ For a = 10, b = 2: (a + b)2 = 144 is greater than a2 + b2 = 104

Common mistakes that cost marks

  • Writing (10 + 2)2 = 102 + 22 = 104. You must add first and then square: 122 = 144.
  • Concluding that (a + b)2 is always bigger after one example. With one negative number it can be smaller.
  • Forgetting that the difference is 2ab, not ab.

How this can come in the exam

MCQ (1 mark)

If a = 7 and b = 3, then (a + b)2 − (a2 + b2) equals

  1. 21
  2. 42
  3. 58
  4. 100
Show answer

(B) 42
The difference is 2ab = 2 × 7 × 3 = 42. (100 − 58 = 42.)

Short answer (2 marks)

Without squaring 13, show that 132 − (92 + 42) = 72.

Show answer13 = 9 + 4, so 132 = (9 + 4)2 = 92 + 2 × 9 × 4 + 42 (1 mark). So 132 − (92 + 42) = 2 × 9 × 4 = 72 (1 mark).

Try one yourself

If a = 6 and b = −5, find (a + b)2 and a2 + b2. Which is greater?

Show answer

(a + b)2 = 12 = 1; a2 + b2 = 36 + 25 = 61. So a2 + b2 is greater, by −2ab = 60.

More questions like this

All Algebraic identities questions · All maths questions

← What is the difference between an equation and an identity?1. What can you say about a and b if (a + b)² < a² + b²? 2. What can y →