Y=-5t^2+8t+1

Solution for Y=-5t^2+8t+1 equation:

=-5Y^2+8Y+1

We move all terms to the left:

-(-5Y^2+8Y+1)=0

We get rid of parentheses

5Y^2-8Y-1=0

a = 5; b = -8; c = -1;
Δ = b2-4ac
Δ = -82-4·5·(-1)
Δ = 84
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{84}=\sqrt{4*21}=\sqrt{4}*\sqrt{21}=2\sqrt{21}$
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{21}}{2*5}=\frac{8-2\sqrt{21}}{10}
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{21}}{2*5}=\frac{8+2\sqrt{21}}{10}
$