Question

If alpha and beta are the zeroes of x^{​​​​​​2}-x-k such that alpha - beta=9 , find the value of k.

Answer 1

Given α and β are the zeroes of the polynomial x2− x -k
Also given that α − β = 9 → (1)
Recall that sum of roots (α + β) = −(b/a)
∴ α + β = 1→ (2)
Add (1) and (2), we get
α − β = 9
α + β = 1
2α = 10
∴ α = 5
Put α = 5 in α + β = 1
5 + β = 1
∴ β = -4
Since 5 and -4 are zeroes of the given polynomial
Put x = 5 in the given polynomial to find the value of k
x2 − x - k
⇒ 25− (5) - k = 0
⇒ 20 - k = 0
⇒
∴ k = 20