Question

If alpha and beeta are the zeros of the quadratic polynomial f(x)=kx 2 +2x-15 such that alpha 2 +beeta 2 =34,then find the value of k

Answer 1

f(x) = kx²+2x-15
Sum of the roots, α + β = $\frac{-b}{a}$

α + β = $\frac{-2}{k}$ 

The product of the roots, αβ = $\frac{c}{a}$
αβ = $\frac{-15}{k}$

Given that

α²+β² = 34
         
(α+β)² - 2αβ = 34

$( \frac{-2}{k})^2 - 2( \frac{-15}{k}) = 34$

$\frac{4}{k^2} + \frac{30}{k} = 34$
Multiplying both sides by k²

4 + 30k = 34k²

34k² - 30k - 4 = 0

17k² - 15k - 2 = 0

(k-1)(17k+2) = 0

k = 1 or -17/2