If a + b = 7 and ab = 6, find $a^2 - b^2$.

Given, a + b = 7 and ab = 6.

We need to find the value of (a - b),

$(a + b)^2 - (a - b)^2 = 4ab\\[1em] ⇒ (a - b)^2 = (a + b)^2 - 4ab$

Substituting the value of (a + b) and ab, we get:

$= (7)^2 - 4\times 6\\[1em] = 49 - 24\\[1em] ⇒ (a - b)^2 = 25\\[1em] ⇒ a - b = \sqrt{25}\\[1em] ⇒ a - b = 5 \text{ or } -5$

We need to find the value of, $a^2 - b^2$ = (a - b)(a + b)

= 5 x 7 or (-5) x 7

= 35 or -35

Hence, $a^2 - b^2$ = 35 or -35.