x^2-5x-14=3x+31

Solution for x^2-5x-14=3x+31 equation:

x^2-5x-14=3x+31

We move all terms to the left:

x^2-5x-14-(3x+31)=0

We get rid of parentheses

x^2-5x-3x-31-14=0

We add all the numbers together, and all the variables

x^2-8x-45=0

a = 1; b = -8; c = -45;
Δ = b2-4ac
Δ = -82-4·1·(-45)
Δ = 244
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:

$\sqrt{\Delta}=\sqrt{244}=\sqrt{4*61}=\sqrt{4}*\sqrt{61}=2\sqrt{61}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{61}}{2*1}=\frac{8-2\sqrt{61}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{61}}{2*1}=\frac{8+2\sqrt{61}}{2}
$