H(t)=-16t2+46t+0.5

Solution for H(t)=-16t2+46t+0.5 equation:

(H)=-16H^2+46H+0.5

We move all terms to the left:

(H)-(-16H^2+46H+0.5)=0

We get rid of parentheses

16H^2-46H+H-0.5=0

We add all the numbers together, and all the variables

16H^2-45H-0.5=0

a = 16; b = -45; c = -0.5;
Δ = b2-4ac
Δ = -452-4·16·(-0.5)
Δ = 2057
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{2057}=\sqrt{121*17}=\sqrt{121}*\sqrt{17}=11\sqrt{17}$
$H_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-45)-11\sqrt{17}}{2*16}=\frac{45-11\sqrt{17}}{32}
$
$H_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-45)+11\sqrt{17}}{2*16}=\frac{45+11\sqrt{17}}{32}
$