Finding the x-intercept of a cubic function

Question

How would one go about solving for x in this cubic equation?

$f(x) = x^3 + x - 4$

I don't think factoring works in this situation:
$f(x) = x^2(x + 1) - 4$

and I don't know how to find the value of $x$ where f(x) = 0.

Answer 1

Alright, define real numbers $u,v$ with, say, $u \geq v,$ then $$ u^3 + v^3 = 4, $$ $$ 3uv = -1$$ so that $$ u^3 v^3 = \frac{-1}{27}$$ This is enough information to solve for $u^3, v^3$ separately. Then $u,v$ separate. Meanwhile, the quantity $u+v$ is the real root of your original cubic.

Answer 2

For solving cubic equations, Cardano's method is not the $\alpha$ and $\omega$.

If you follow the (simple) steps described here, you will immediately notice that there is only one real root since $\Delta=-436$.

So, use the hyperbolic method and obtain $$x=\frac{2}{\sqrt{3}}\sinh \left(\frac{1}{3} \sinh ^{-1}\left(6 \sqrt{3}\right)\right)$$ which is a bit nicer than $$x=\frac{\sqrt[3]{18+\sqrt{327}}}{3^{2/3}}-\frac{1}{\sqrt[3]{3 \left(18+\sqrt{327}\right)}}$$