Let-f-x-x-3-3x-b-and-g-x-x-2-bx-3-where-b-is-a-real-number-What-is-the-sum-of-all-possible-values-of-b-for-which-the-equations-f-x-0-and-g-x-0-have-a-common-root-

Question Number 19698 by Tinkutara last updated on 14/Aug/17

Let f (x) = x^{3} - 3 x + b and g (x) = x^{2} +

b x - 3, where b is a real number . What

is the sum of all possible values of b for

which the equations f (x) = 0 and g (x)

= 0 have a common root ?

Answered by ajfour last updated on 14/Aug/17

let α be the common root .

f (α) = α^{3} - 3 α + b = 0

and α g (α) = α (α^{2} + α b - 3) = 0

subtracting the two we get :

α^{2} b + 3 α - 3 α - b = 0

\Rightarrow α^{2} = 1 or α = \pm 1

as α^{2} + α b - 3 = 0

for α = 1, 1 + b - 3 = 0 \Rightarrow b = 2

for α = - 1, 1 - b - 3 = 0 \Rightarrow b = - 2

Σ b_{i} = 0 .

Commented by Tinkutara last updated on 15/Aug/17

Thank you very much Sir!