31=-8t^2+32t+1

Solution for 31=-8t^2+32t+1 equation:

31=-8t^2+32t+1

We move all terms to the left:

31-(-8t^2+32t+1)=0

We get rid of parentheses

8t^2-32t-1+31=0

We add all the numbers together, and all the variables

8t^2-32t+30=0

a = 8; b = -32; c = +30;
Δ = b2-4ac
Δ = -322-4·8·30
Δ = 64
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{64}=8$
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-32)-8}{2*8}=\frac{24}{16}
=1+1/2
$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-32)+8}{2*8}=\frac{40}{16}
=2+1/2
$