Question

If α, β are zeroes of the polynomial 8x² – 5x – 1, then form a quadratic polynomial in x whose zeroes are `2/α` and `2/β`.

Answer 1

Given α, β are zeroes of 8x² – 5x – 1.

∴ Sum of zeroes (α + β) = `(-(-5))/8`

= `5/8`

Product of zeroes (αβ) = `c/a = (-1)/8`

Quadratic equation whose roots are `2/α` and `2/β` is:

`x^2 - (2/α + 2/β)x + 2/α xx 2/β`

Quadratic equation is

⇒ `x^2 - 2x (1/α + 1/β) + 4/(αβ)`   ...[∵ x² – (sum of zeroes)x + (product of zeroes)]

⇒ `x^2 - 2x ((α + β)/(αβ)) + 4/(αβ)`

⇒ `x^2 - 2x ((5/8)/((-1)/8)) + 4/((-1)/8)`

⇒ x² + 10x – 32