Question

if 2x²-5x+3 then find (i)alpha²+beta². (ii) 1 alpha /2 +1/2beta

Answer 1

Let alpha and beta are the roots of 2x² -5x +3 .
sum of roots = -( coefficient of x)/coefficient of x²
alpha + beta = -(-5)/2 = 5/2

again,
Product of roots = constant/coefficient of x²
alpha.beta = 3/2

(1) alpha² + beta² = (alpha+beta)² -2alpha.beta
= (5/2)² -2×3/2
=25/4 - 12/4
=13/4

(2)1/2.alpha + 1/2.beta
=(alpha + beta)/2alpha.beta
=(5/2)/2×3/2 = 5/6 or, if you mean (2) alpha/2 + beta/2 =(alpha + beta)/2 =(5/2)/2 =5/4