31=x+5x^2

Solution for 31=x+5x^2 equation:

31=x+5x^2

We move all terms to the left:

31-(x+5x^2)=0

We get rid of parentheses

-5x^2-x+31=0

We add all the numbers together, and all the variables

-5x^2-1x+31=0

a = -5; b = -1; c = +31;
Δ = b2-4ac
Δ = -12-4·(-5)·31
Δ = 621
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{621}=\sqrt{9*69}=\sqrt{9}*\sqrt{69}=3\sqrt{69}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-3\sqrt{69}}{2*-5}=\frac{1-3\sqrt{69}}{-10}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+3\sqrt{69}}{2*-5}=\frac{1+3\sqrt{69}}{-10}
$