Question

Solve the following equations:
$x^4-10x^2-20x-16=0$.

Answer 1

Given equation, $x^4-10x^2-20x-16=0$

Consider $f\left(x\right)=x^4-10x^2-20x-16$

By inspection we can easily see that $f(-2)=0$. Therefore, $(x+2)$ is a factor of the given equation

$\therefore f\left(x\right)=(x+2)\cdot g\left(x\right)⟹g\left(x\right)=x^3-2x^2-6x-8$

We need to find root of $g\left(x\right)=0$

$⟹x^3–2x^2-6x-8=0⟹x^3+2x^2+2x-4x^2-8x-8=0⟹(x-4)(x^2+2x+2)=0$

Solving the quadratic we have, $x=-1\pm i$

$\therefore$ the roots of the given equation are $x=-2,4,-1\pm i$