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

Question

Mathematics

Regina

21 July, 01:53

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

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Answers (1)

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Aidan · Answer 1

in steps

Step-by-step explanation:

The question did not state if alpha>beta or alpha
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