Y=0.5x^2-x-2

Solution for Y=0.5x^2-x-2 equation:

=0.5Y^2-Y-2

We move all terms to the left:

-(0.5Y^2-Y-2)=0

We get rid of parentheses

-0.5Y^2+Y+2=0

a = -0.5; b = 1; c = +2;
Δ = b2-4ac
Δ = 12-4·(-0.5)·2
Δ = 5
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

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