Question

If a² + b² = 65 and ab = 8, find the value of a² − b².

Answer 1

Here, a² + b² = 65 and ab = 8,

(1) Using the identity for (a + b)².

(a + b)² = a² + 2ab + b²

Substitute the values:

(a + b)² = 65 + 2 × 8

∴ (a + b)² = 81

∴ a + b = `+-sqrt81`

∴ a + b = ±9

(2) Using the identity for (a − b)².

(a − b)² = a² − 2ab + b²

Substitute the values:

(a − b)² = 65 − 2 × 8

∴ (a − b)² = 65 − 16

∴ (a − b)² = 49

∴ a − b = `+-sqrt49`

∴ a − b = ±7

(3) Thus, calculating a² − b² using the difference of squares formula:

a² − b² = (a − b) (a + b)

Substitute the values:

a² − b² = (±7) (±9)

∴ a² − b² = ±63

Hence, the value of a² − b² is ± 63.