200p-0.5p^2=400

Solution for 200p-0.5p^2=400 equation:

200p-0.5p^2=400

We move all terms to the left:

200p-0.5p^2-(400)=0

a = -0.5; b = 200; c = -400;
Δ = b2-4ac
Δ = 2002-4·(-0.5)·(-400)
Δ = 39200
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$p_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$p_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{39200}=\sqrt{19600*2}=\sqrt{19600}*\sqrt{2}=140\sqrt{2}$
$p_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(200)-140\sqrt{2}}{2*-0.5}=\frac{-200-140\sqrt{2}}{-1}
$
$p_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(200)+140\sqrt{2}}{2*-0.5}=\frac{-200+140\sqrt{2}}{-1}
$