x^2+x=7.75

Solution for x^2+x=7.75 equation:

x^2+x=7.75

We move all terms to the left:

x^2+x-(7.75)=0

We add all the numbers together, and all the variables

x^2+x-7.75=0

a = 1; b = 1; c = -7.75;
Δ = b2-4ac
Δ = 12-4·1·(-7.75)
Δ = 32
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:

$\sqrt{\Delta}=\sqrt{32}=\sqrt{16*2}=\sqrt{16}*\sqrt{2}=4\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-4\sqrt{2}}{2*1}=\frac{-1-4\sqrt{2}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+4\sqrt{2}}{2*1}=\frac{-1+4\sqrt{2}}{2}
$