If (alpha) and (beta) are the roots of (x^{2} + x – 2 = 0), find the value of (frac{1}{alpha^{2}} + frac{1}{beta^{2}}).
-
A.
(frac{5}{4}) -
B.
(frac{3}{4}) -
C.
(frac{1}{4}) -
D.
(frac{-3}{4})
Correct Answer: Option A
Explanation
Given, (x^{2} + x – 2 = 0), a = 1, b = 1 and c = -2.
(alpha + beta = frac{-b}{a} = frac{-1}{1} = -1)
(alphabeta = frac{c}{a} = frac{-2}{1} = -2)
(frac{1}{alpha^{2}} + frac{1}{beta^{2}} = frac{beta^{2} + alpha^{2}}{(alphabeta)^{2}})
(beta^{2} + alpha^{2} = (alpha + beta)^{2} – 2alphabeta = (-1)^{2} – 2(-2) = 1 + 4 = 5)
(frac{1}{alpha^{2}} + frac{1}{beta^{2}} = frac{5}{(-2)^{2}} = frac{5}{4}).