0.5x^2=2x+1

Solution for 0.5x^2=2x+1 equation:

0.5x^2=2x+1

We move all terms to the left:

0.5x^2-(2x+1)=0

We get rid of parentheses

0.5x^2-2x-1=0

a = 0.5; b = -2; c = -1;
Δ = b2-4ac
Δ = -22-4·0.5·(-1)
Δ = 6
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{-(-2)-\sqrt{6}}{2*0.5}=\frac{2-\sqrt{6}}{1}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+\sqrt{6}}{2*0.5}=\frac{2+\sqrt{6}}{1}
$