.5x-1=5-x^2

Solution for .5x-1=5-x^2 equation:

.5x-1=5-x^2

We move all terms to the left:

.5x-1-(5-x^2)=0

We get rid of parentheses

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

We add all the numbers together, and all the variables

x^2+.5x-6=0

a = 1; b = .5; c = -6;
Δ = b2-4ac
Δ = .52-4·1·(-6)
Δ = 24.25
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(.5)-\sqrt{24.25}}{2*1}=\frac{-0.5-\sqrt{24.25}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(.5)+\sqrt{24.25}}{2*1}=\frac{-0.5+\sqrt{24.25}}{2}
$