65=4.5t^2+43.5t+17

Solution for 65=4.5t^2+43.5t+17 equation:

65=4.5t^2+43.5t+17

We move all terms to the left:

65-(4.5t^2+43.5t+17)=0

We get rid of parentheses

-4.5t^2-43.5t-17+65=0

We add all the numbers together, and all the variables

-4.5t^2-43.5t+48=0

a = -4.5; b = -43.5; c = +48;
Δ = b2-4ac
Δ = -43.52-4·(-4.5)·48
Δ = 2756.25
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-43.5)-\sqrt{2756.25}}{2*-4.5}=\frac{43.5-\sqrt{2756.25}}{-9}
$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-43.5)+\sqrt{2756.25}}{2*-4.5}=\frac{43.5+\sqrt{2756.25}}{-9}
$