Question

If one root of $x^3+a{x}^2+bx+c=0$ is the sum of the other two roots, then

• A. $a^3=4(ab-c)$
• B. $a^3=4(ab-2c)$
• C. $a^3=ab-c$
• D. $a^3=ab-2c$

Answer 1

The correct option is B $a^3=4(ab-2c)$

Let the roots be $\alpha ,\beta ,\gamma$
Then
$\alpha =\beta +\gamma$.
Hence
$\alpha +\beta +\gamma =-a$
$2(\beta +\gamma )=-a$
$\beta +\gamma =\alpha =\frac{-a}{2}$ ...(i)
$\alpha .\beta +\beta .\gamma +\gamma .\alpha =b$
$\alpha (\beta +\gamma )+\beta .\gamma =b$
${\alpha }^2+\beta .\gamma =b$
Or
$\frac{a^2}{4}+\beta .\gamma =b$
$a^2+4\beta .\gamma =4b$ ...(ii)
And
$\alpha .\beta .\gamma =-c$
Or
$\frac{-a}{2}.\beta .\gamma =-c$
Or
$\beta .\gamma =\frac{2c}{a}$.
Then
$a^2+4\beta .\gamma =4b$
$a^2+4\frac{2c}{a}=4b$
$a^3+8c=4ab$
Or
$a^3=4(ab-2c)$.