If a + b = 10 and a² + b² = 68, find the value of ab.

If $a + b = 10$ and $a^{2} + b^{2} = 68$ , find the value of $ab$ .

Class 8Algebraic Expressions and IdentitiesAlgebra

(a + b)^{2} = a^{2} + 2 ab + b^{2}

, so

1 0^{2} = 68 + 2 ab

, giving

2 ab = 100 - 68 = 32

and

ab = 16

. (Check with

a = 8, b = 2

: sum 10, squares 64 + 4 = 68, product 16.) Stopping at

2 ab = 32

is the commonest slip; 34 halves 68 instead; 84 averages 100 and 68.

Answer: D) 16

Identities save time and reveal structure. Examples of clever uses:

- Compute

10 2^{2}

mentally:

(100 + 2)^{2} = 10 0^{2} + 2 (100) (2) + 2^{2} = 10000 + 400 + 4 = 10404

.
- Compute

99 \times 101 : (100 - 1) (100 + 1) = 10 0^{2} - 1^{2} = 10000 - 1 = 9999

.
- Compute

4 7^{2} - 4 3^{2} : (47 - 43) (47 + 43) = 4 \times 90 = 360

.

Look at any number-pair calculation; if you can rewrite it as one of the three identity forms, the answer falls out.

Don't just memorise the answer — build a daily practice habit.