34=16t^2+80t+5

Solution for 34=16t^2+80t+5 equation:

34=16t^2+80t+5

We move all terms to the left:

34-(16t^2+80t+5)=0

We get rid of parentheses

-16t^2-80t-5+34=0

We add all the numbers together, and all the variables

-16t^2-80t+29=0

a = -16; b = -80; c = +29;
Δ = b2-4ac
Δ = -802-4·(-16)·29
Δ = 8256
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:

$\sqrt{\Delta}=\sqrt{8256}=\sqrt{64*129}=\sqrt{64}*\sqrt{129}=8\sqrt{129}$
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-80)-8\sqrt{129}}{2*-16}=\frac{80-8\sqrt{129}}{-32}
$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-80)+8\sqrt{129}}{2*-16}=\frac{80+8\sqrt{129}}{-32}
$