4q^2-8q=31

Solution for 4q^2-8q=31 equation:

4q^2-8q=31

We move all terms to the left:

4q^2-8q-(31)=0

a = 4; b = -8; c = -31;
Δ = b2-4ac
Δ = -82-4·4·(-31)
Δ = 560
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{560}=\sqrt{16*35}=\sqrt{16}*\sqrt{35}=4\sqrt{35}$
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{35}}{2*4}=\frac{8-4\sqrt{35}}{8}
$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{35}}{2*4}=\frac{8+4\sqrt{35}}{8}
$