Question

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. 

`1/4 , -1`

Answer 1

Given: α + β = `1/4`, αβ = -1

Since ax² + bx + c = kx² - k(α + β)x + kαβ

In comparison,

a = k, b = -k(α + β) and c = kαβ

α + β = `(-b)/a = 1/4` and αβ = `c/a = -1`

⇒ a = 4

⇒ b = -4(α + β) 

⇒ c = kαβ = 4(-1)

Hence, on writing as ax² + bx + c

⇒ 4x² - 4(α + β)x + 4(αβ)

⇒ `4x^2 - 4(1/4)x + 4(-1)`

⇒ 4x² - x - 4

The quadratic polynomial is 4x² - x - 4.