Question

Find the solutions of the following equations which have common roots:
$4x^4+12x^3-x^2-15x=0,6x^4+13x^3-4x^2-15x=0$.

Answer 1

Let $f\left(x\right)=4x^4+12x^3-x^2-15x\text{and}g\left(x\right)=6x^4+13x^3-4x^2-15x$

H.C.F. of $f\left(x\right)$ and $g\left(x\right)$ is $x(x-1)(2x-3)$

Dividing $f\left(x\right)$ and $g\left(x\right)$ by $x(x-1)(2x-3)$, we get

$f\left(x\right)=x(x-1)(2x-3)(2x+5)\text{and}g\left(x\right)=x(x-1)(2x-3)(3x+5)$

$\therefore$ the roots of $f\left(x\right)=0$ are $0,1,-\frac{3}{2},-\frac{5}{2}$

and the roots of $g\left(x\right)=0$ are $0,1,-\frac{3}{2},-\frac{5}{3}$