288=74+48t+16t^2

Solution for 288=74+48t+16t^2 equation:

288=74+48t+16t^2

We move all terms to the left:

288-(74+48t+16t^2)=0

We get rid of parentheses

-16t^2-48t-74+288=0

We add all the numbers together, and all the variables

-16t^2-48t+214=0

a = -16; b = -48; c = +214;
Δ = b2-4ac
Δ = -482-4·(-16)·214
Δ = 16000
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:

$\sqrt{\Delta}=\sqrt{16000}=\sqrt{1600*10}=\sqrt{1600}*\sqrt{10}=40\sqrt{10}$
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-48)-40\sqrt{10}}{2*-16}=\frac{48-40\sqrt{10}}{-32}
$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-48)+40\sqrt{10}}{2*-16}=\frac{48+40\sqrt{10}}{-32}
$