Explanation:
the target height (ground) is 0 ft, I assume.
the rocket starts from the height of the tower.
FYI - this we get for x = 0 seconds, which calculates the height of the rocket before it even starts.
y = -16x² + 119x + 57
for x = 0 we get y = 57 ft
anyway, we need to solve
0 = - 16x² + 119x + 57
the general solution for a quadratic equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = -16
b = 119
c = 57
x = (-119 ± sqrt(119² - 4×-16×57))/(2×-16) =
= (-119 ± sqrt(14161 + 3648))/-32 =
= (-119 ± sqrt(17809))/-32
x1 = (-119 + 133.4503653...)/-32 = -0.451573916...
x2 = (-119 - 133.4503653...)/-32 = 7.889073916...
the negative solution as time does not make sense (this would be the time calculated back to the ground at the start), so x2 is our only solution :
the rocket will hit the ground after 7.89 seconds.