-x^2+8=x-3

Solution for -x^2+8=x-3 equation:

-x^2+8=x-3

We move all terms to the left:

-x^2+8-(x-3)=0

We add all the numbers together, and all the variables

-1x^2-(x-3)+8=0

We get rid of parentheses

-1x^2-x+3+8=0

We add all the numbers together, and all the variables

-1x^2-1x+11=0

a = -1; b = -1; c = +11;
Δ = b2-4ac
Δ = -12-4·(-1)·11
Δ = 45
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:

$\sqrt{\Delta}=\sqrt{45}=\sqrt{9*5}=\sqrt{9}*\sqrt{5}=3\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-3\sqrt{5}}{2*-1}=\frac{1-3\sqrt{5}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+3\sqrt{5}}{2*-1}=\frac{1+3\sqrt{5}}{-2}
$