Step by step
The question
Question: A ball is thrown up from the ground.
Its height after t seconds is h = -16t² + 80t feet.
When does it land?
1 Find what we know
What do we know?
The height formula: h = −16t² + 80t feet, after t seconds.
It lands when it is back on the ground: the height is 0.
What do we want?
The time when it lands.
2 Name the unknown
The formula already has a letter for the time: t, in seconds.
3 Write the equation
In words: the height is 0.
So the equation is −16t² + 80t = 0.
There is t times t in it: a quadratic equation.
Now we solve it.
The t² part is negative.
That is harder to work with.
So we multiply both sides by −1.
Every sign changes, and −1 times 0 is still 0.
4 Divide by 16
Easy way first: every number can be divided by 16.
So we divide both sides by 16.
0 divided by 16 is still 0.
Now we have t² − 5t = 0.
5 Factor
There is no plain number.
Both parts have t in them.
t² is t times t, and 5t is t times −5.
So we take t out: t times (t − 5).
So it is t(t − 5) = 0.
6 One bracket is 0
t times the bracket is 0.
That only happens if one of them is 0.
We make each part 0.
Two small equations.
t = 0 is done already.
In t − 5 = 0, we add 5 on both sides.
So t = 0 or t = 5.
7 Answer the question
t = 0: that is the moment it is thrown.
So it is not the answer.
Careful!
At t = 0 the height is 0 too.
But that is when it is thrown, not when it lands.
t = 5 works: it lands.
It lands after 5 seconds.
8 Check
Let's check in the story.
Put t = 5 into the height.
We get 0, on the ground.
So the answer is right!
9 Type it in
The answer box takes 5 characters.
5 has 1.
It fits.
We type 5.
Let's look back.
First: find what we know.
Then: name the unknown.
Then: write the equation.
Then: solve it.
Then: answer the question.
Last: check.