Equation Geeks!

$x^3+x=-30 ... ...(1)$

$x^2+x=6 ........(2)$

From $(1):$

$\begin{aligned} x^3 + x & = -30 \\x^3 + x + 30 & = 0 \\ (x+3)(x^2 - 3x + 10) & = 0 \\\implies x &= \color{#3D99F6} -3 \\ \end{aligned}$

From $(2):$

$\begin{aligned} x^2 + x &= 6 \\ x^2 + x - 6 & = 0 \\ x^2 + 3x - 2x - 6 & = 0 \\ x(x + 3) - 2(x + 3) & = 0\\(x-2)(x+3) & = 0 \\ \end{aligned}$

$\implies x = 2$ and $x = \color{#3D99F6} -3$ .

The final solution is $x = \color{#3D99F6} \boxed{-3}$ -

From: x² + x = 6 x² + x - 6 = 0 (x + 3)(x - 2) = 0 x = -3 or x = 2

Let's check for each of the solution, to x³ + x = -30

If x = -3, (-3)³ + (-3) = -30 -27 - 3 = -30 -30 = -30

If x = 2, 2³ + 2 = -30 8 + 2 = -30 10 = 30 (?) (It's not satisfied the equation)

So, the only solution is x = -3