Question

alpha, beta, gamma are the roots of the equation x^3-10x^2+7x+8=0 then alpha +beta+gamma=? ​

Answer 1

To find:

$\sf{The \ value \ of \ \alpha+\beta+\gamma}$

Solution:

$\sf{The \ given \ cubic \ equation \ is}$

$\sf{\longmapsto{x^{3}-10x^{2}+7x+8=0}}$

$\sf{Here, \ a=1, \ b=-10, \ c=7 \ and \ d=8}$

Relationship between coefficients and roots.

$\sf{\alpha+\beta+\gamma=-\dfrac{b}{a}}$

$\sf{\alpha\beta+\gamma\beta+\gamma\alpha=\dfrac{c}{a}}$

$\sf{\alpha\beta\gamma=-\dfrac{d}{a}}$

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$\sf{we \ know,}$

$\sf{\alpha+\beta+\gamma=-\dfrac{b}{a}}$

$\sf{Substituting \ values \ of \ b \ and \ a, \ we \ get}$

$\sf{\alpha+\beta+\gamma=-\dfrac{-10}{1}}$

$\sf{\therefore{\alpha+\beta+\gamma=10}}$

$\sf\purple{\tt{\therefore{The \ value \ of \ \alpha+\beta+\gamma \ is \ 10.}}}$