500=50t+5t^2

Solution for 500=50t+5t^2 equation:

500=50t+5t^2

We move all terms to the left:

500-(50t+5t^2)=0

We get rid of parentheses

-5t^2-50t+500=0

a = -5; b = -50; c = +500;
Δ = b2-4ac
Δ = -502-4·(-5)·500
Δ = 12500
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{12500}=\sqrt{2500*5}=\sqrt{2500}*\sqrt{5}=50\sqrt{5}$
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-50)-50\sqrt{5}}{2*-5}=\frac{50-50\sqrt{5}}{-10}
$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-50)+50\sqrt{5}}{2*-5}=\frac{50+50\sqrt{5}}{-10}
$