Question

If alpha and beta are the roots of the equation x^2+kx+12=0 and alpha - beta =1, find k.

Answer 1

let alpha be n and beta be m

here a = 1, b = k, c = 12

n - m = 1 .: n = 1 + m .......(1)

n + m = - k

substituting (1) 1 + m + m = - k or 2m + 1 = - k
m = (- k - 1) / 2 .....(2)

nm = 12 substituting (1) (m + 1) m = 12 or m2 + m = 12 ....(3)

from (2) and (3),

[(- k -1) / 2 ]2 + (- k - 1) / 2 = 12

(- k - 1)2 / 4 + (- k - 1) / 2 = 12

(- k - 1)2 + 2 (- k - 1) = 48

k2 + 2k + 1 - 2k - 2 = 48

k2 - 1 = 48

k2 = 49

K=±7

Answer 2

Let the two roots be α and β
Given α : β = 1 : 3
That is α / β = 1 / 3
β = 3 α
Given quadratic equation is x2 +kx+12=0
Sum of roots (α + β) = -b/a = - k
That is (α + 3 α) = - k
Hence α = - k/4 ---- (1)
Product of roots, (αβ) = c/a = 12/1
That is (3α2) = 12
α2 = 4
α = ± 2
Equation (1) becomes,
2 = - k/4
Hence k = -8
If α = - 2
k = 8