Question

Find the solutions of the following equations which have common roots:
$2x^4-2x^3+x^2+3x-6=0,4x^4-2x^3+3x-9=0$.

Answer 1

Let $f\left(x\right)=2x^4-2x^3+x^2+3x-6\text{and}g\left(x\right)=4x^4-2x^3+3x-9$

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

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

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

$\therefore$ the roots of $f\left(x\right)=0$ are $\pm \sqrt{\frac{3}{2}},\frac{1\pm \sqrt{-7}}{2}$

and the roots of $g\left(x\right)=0$ are $\pm \sqrt{\frac{3}{2}},\frac{1\pm \sqrt{-23}}{4}$