65=2x+0.5x^2

Solution for 65=2x+0.5x^2 equation:

65=2x+0.5x^2

We move all terms to the left:

65-(2x+0.5x^2)=0

We get rid of parentheses

-0.5x^2-2x+65=0

a = -0.5; b = -2; c = +65;
Δ = b2-4ac
Δ = -22-4·(-0.5)·65
Δ = 134
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{134}}{2*-0.5}=\frac{2-\sqrt{134}}{-1}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+\sqrt{134}}{2*-0.5}=\frac{2+\sqrt{134}}{-1}
$