Back to the Ground

A ball kicked up from the ground has height h = -5t^2 + 10t metres after t seconds. When does it land? Landing means height 0, so solve -5t^2 + 10t = 0.

There is no constant term, so both terms share the factor t, and -5 as well. Taking out -5t gives -5t(t - 2) = 0. A product is 0 only when one of its factors is 0, so split into cases: t = 0 or t - 2 = 0. The answers are 0 and 2 seconds.

Both answers are real moments. At t = 0 the ball is on the ground because it has just been kicked; at t = 2 it lands. The top of the flight is halfway between them, at t = 1, because a parabola is symmetric about its vertex.