Question

If equations $x^6+a{x}^4-4x=0$ and $x^7+a{x}^5-1=0$ have a common root, then which of the following statements is (are) CORRECT ?

• A. Sum of all possible values of $a$ is $\frac{-1}{2}.$
• B. Sum of all possible common roots is $0.$
• C. Sum of all possible common roots is $1.$
• D. Sum of all possible values of $a$ is $0.$

Answer 1

Answer: (A), (B)

Sum of all possible common roots is $0.$

$x^7+a{x}^5-1=0...\left(1\right)$
Clearly, eqn. $\left(1\right)$ does not have $x=0$ as a root.
And $x^6+a{x}^4-4x=0$
So, multiplying by $x,$
$x^7+a{x}^5-4x^2=0...\left(2\right)$

Equation $\left(1\right)-\left(2\right),$
$4x^2=1$
$\Rightarrow x=\pm \frac{1}{2}$
Sum of all possible common roots $=0.$

If $x=\frac{1}{2}$ is a solution,
then ${\left(\frac{1}{2}\right)}^7+a{\left(\frac{1}{2}\right)}^5-1=0$
$\Rightarrow 1+4a-128=0$
$\Rightarrow a=\frac{127}{4}$

If $x=\frac{-1}{2}$ is a solution,
then ${\left(\frac{-1}{2}\right)}^7+a{\left(\frac{-1}{2}\right)}^5-1=0$
$\Rightarrow a=\frac{-129}{4}$
Sum of possible values of $a$ is $\frac{-1}{2}$.