Question

Let p and q be two positive numbers such that p + q = 2 and p⁴ + q⁴ = 272. Then p and q are roots of the equation ______.

Options

• x² – 2x + 2 = 0

• x² – 2x + 8 = 0

• x² – 2x + 136 = 0

• x² – 2x + 16 = 0

Answer 1

Let p and q be two positive numbers such that p + q = 2 and p⁴ + q⁴ = 272. Then p and q are roots of the equation `underlinebb(x^2 - 2x + 16 = 0)`.

Explanation:

Given: p and q are two positive numbers such that p + q = 2 and p⁴ + q⁴ = 272

(p + q)² = 2²

⇒ p² + q² + 2pq = 4

p² + q² = 4 – 2pq

Squaring both sides, 

(p² + q²)² = (4 – 2pq)²

⇒ p⁴ + q⁴  + 2p²q² = 16 + 4p²q² – 16pq

⇒ 272 – 2p²q² = 16 + 4p²q² – 16pq

⇒ 272 – 2p²q² = 16 – 16pq  ...(∵ p⁴ + q⁴ = 272)

⇒ p²q² – 8pq – 128 = 0

⇒ (pq)² – 8pq – 128 = 0

pq = `(8 +- 24)/2` = 16 or –8

⇒ pq = 16

Now, quadratic equation whose roots are p, q is x² – (p + q)x + pq = 0

 ⇒ x² – 2x + 16 = 0