Find-the-sum-of-the-real-roots-of-equations-a-3-6a-2-15a-17-0-and-a-3-6a-2-15a-11-0-

Question Number 152678 by mathdanisur last updated on 31/Aug/21

Find the sum of the real roots of equations :

a^{3} - {6 a}^{2} + 15 a - 17 = 0 and

a^{3} - {6 a}^{2} + 15 a - 11 = 0

Answered by mr W last updated on 31/Aug/21

e q n . 1 :

s a y r o o t i s a = 2 + t

{(2 + t)}^{3} - 6 {(2 + t)}^{2} + 15 (2 + t) - 17 = 0

\Rightarrow t^{3} + 3 t - 3 = 0 \dots (i)

e q n . 2 :

s a y r o o t i s b = 2 + s

{(2 + s)}^{3} - 6 {(2 + s)}^{2} + 15 (2 + s) - 11 = 0

\Rightarrow s^{3} + 3 s + 3 = 0

\Rightarrow {(- s)}^{3} + 3 (- s) - 3 = 0 \dots (i i)

f r o m (i) a n d (i i) :

t = - s

s u m o f r e a l r o o t s o f e q n . 1 a n d 2 :

a + b = (2 + t) + (2 + s) = 4 + t - t = 4

Commented by mathdanisur last updated on 01/Sep/21

Thank you Ser

Answered by MJS_new last updated on 31/Aug/21

(1) a = 2 + u^{1 / 3} + v^{1 / 3}

(2) a = 2 - u^{1 / 3} - v^{1 / 3}

\Rightarrow answer is 4

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