Here's our equation.
[tex]h=176-16t^2[/tex]
Let's reorder this so t² comes before t.
[tex]h=-16t^2+176t[/tex]
We want to find out when the ball hits the ground. Essentially, what is t when h=0?
Let's plug in 0 for h and solve.
[tex]0=-16t^2+176t[/tex]
From here we can use a number of methods to solve.
Methods such as splitting the middle, quadratic formula, etc. are not necessary for this equation, however. The best method here is just plain factoring.
-16t² and 176t are both divisible by 16t.
[tex]0=16t(-t+11)[/tex]
Now we want t to equal zero. If either of those factors were zero, then the whole thing would be zero when we multiply by it. The first option is obvious. If t were 0, 16t would be 0. That's when the ball is first kicked off the ground. The other, 11-t, works out to be t = 11.
