Question

if a, b are the roots of 3x²-2x+4=0
then find
a³+b³=?
​

Answer 1

Step-by-step explanation:

(x - alpha)(x - beta) = 0

x^2 - (alpha + beta)x + (alpha)(beta) = 0

3x^2 - 3(alpha + beta)x + 3(alpha)(beta) = 0

By correspondence,

3(alpha + beta) = 2, so alpha + beta = 2/3

3(alpha)(beta) = 4, so (alpha)(beta) = 4/3

(alpha + beta)^2 = (2/3)^2 = 4/9

(alpha)^2 + (beta)^2 + 2(alpha)(beta) = 4/9

(alpha)^2 + (beta)^2 + 2 (4/3) = 4/9

(alpha)^2 + (beta)^2 = 4/9 - 8/3 = 4/9 - 24/9

= -20/9

(alpha/beta)+ (beta/alpha)

= (alpha^2 + beta^2)/(alpha * beta)

= (-20/9)/(4/3) * (9/9)

= (-20)/(12)

= -5/3