If p - q = 9 and pq = 36, evaluate

(i) p + q

(ii) p² - q².

(i) We know that,

(p - q)² = p² + q² - 2pq

⇒ (p - q)² = p² + q² + 2pq - 2pq - 2pq

⇒ (p - q)² = (p + q)² - 4pq

⇒ (p + q)² = (p - q)² + 4pq

⇒ (p + q) = $\sqrt{(p - q)^2 + 4pq}$

Substituting value we get,

$\Rightarrow p + q = \sqrt{9^2 + 4 \times 36} \\[1em] \Rightarrow p + q = \sqrt{81 + 144} \\[1em] \Rightarrow p + q = \sqrt{225} \\[1em] \Rightarrow p + q = \pm 15.$

Hence, p + q = ±15.

(ii) p² - q² = (p - q)(p + q).

Substituting value we get,

⇒ p² - q² = 9 × ±15 = ±135.

Hence, p² - q² = ±135.