Question

Solve the following equation that has equal roots:
$8x^4+4x^3-18x^2+11x-2=0$.

Answer 1

Given equation is : $8x^4+4x^3-18x^2+11x-2=0$

Consider $f\left(x\right)=8x^4+4x^3-18x^2+11x-2$

$\therefore f^{\prime }\left(x\right)=32x^3+12x^2-36x+11$

Now, HCF of $f\left(x\right)$ and $f^{\prime }\left(x\right)$ is $(x-\frac{1}{2})$. Hence $\frac{1}{2}$ is a double root of $f\left(x\right)=0$

$f\left(x\right)$ can be factored as $4(x-\frac{1}{2}{)}^2(2x^2+3x-2\left)\text{or}4\right(x-\frac{1}{2}{)}^2(x+2)(2x-1)$

$\therefore$ roots of the given equation are $\frac{1}{2},\frac{1}{2},\frac{1}{2},-2$