If alpha and beta are the roots of the equation 3x^2-9x+2=0 find the values of: (I) alpha ×beta + alpha^2 × beta. (ii) alpha^2-alpha×beta+beta^2

in steps

Step-by-step explanation:

The question did not state if alpha>beta or alpha<beta, so the answer will have 2 answers  for each questions

3x²-9x+2=0

x = (-(-9) ± √(-9)²-4*(3)*(2)) / (2*3)

x = (9 + √57) / 6   or x = (9 - √57) / 6      (alpha and beta) or (beta and alpha)

(I) alpha (a) ×beta (b) + alpha² × beta = ab (1+a)

   = ((9 + √57) / 6) ((9 - √57) / 6) (1 + (9 ± √57))  

   = ((9² - (√57)²)/36) (10 ± √57)

   = (24/36) (10 ± √57)

   = 2/3 (10 ± √57)   or (11.7 or 1.63)

(ii) alpha²-alpha×beta+beta² = a² -2ab + b² +ab = (a - b)² + ab

if a is alpha    

= ((9 + √57) / 6) - ((9 - √57) / 6)) + ((9 + √57) / 6) ((9 - √57) / 6))  

    = √57/3 + 2/3

     = (√57 + 2) / 3

if a is beta

((9 - √57) / 6) - ((9 + √57) / 6)) + ((9 - √57) / 6) ((9 + √57) / 6))  

    = - √57/3 + 2/3

     = - (√57 + 2) / 3