If a + b = 1 and a - b = 7; find:

(i) a² + b²

(ii) ab

(i) Given: a + b = 1 and a - b = 7

Squaring the given equations, we get:

⇒ (a + b)² = 1²

⇒ a² + b² + 2ab = 1 ……………….(1)

⇒ (a - b)² = 7²

⇒ a² + b² - 2ab = 49 ……………….(2)

Adding equation (1) and (2),

⇒ (a² + b² + 2ab) + (a² + b² - 2ab) = 1 + 49

⇒ a² + b² + 2ab + a² + b² - 2ab = 50

⇒ 2a² + 2b² = 50

⇒ 2(a² + b²) = 50

⇒ a² + b² = $\dfrac{50}{2}$

⇒ a² + b² = 25

Hence, the value of a² + b² = 25.

(ii) Subtracting equation (2) from equation (1),

⇒ (a² + b² + 2ab) - (a² + b² - 2ab) = 1 - 49

⇒ a² + b² + 2ab - a² - b² + 2ab = -48

⇒ 4ab = -48

⇒ ab = $\dfrac{-48}{4}$

= -12

Hence, the value of ab = -12.