# How do you solve x (x^2 + 2x + 3)=4?

May 22, 2015

Distribute: ${x}^{3} + 2 {x}^{2} + 3 x = 4$

Rearrange: ${x}^{3} + 2 {x}^{2} + 3 x - 4 = 0$

It would be great if we could now factor the left-hand side. Unfortunately, the roots of this are an irrational real number as well as two complex numbers with irrational real and imaginary parts.

Using a computer or calculator gives approximate roots:

$x \setminus \approx 0.776045$, $x \setminus \approx - 1.38802 \setminus \pm 1.79659 i$

There is a method/formula for finding exact expressions for these roots (involving roots that are irrational), but it's not pleasant:
general method/formula for cubic roots on Wikipedia