Step by step
The question
Question: A ball is thrown up from a cliff.
Its height after t seconds is h = -16t^2 + 32t + 48 feet.
When does it hit the ground?
1 Find what we know
What do we know?
The height formula: h = −16t² + 32t + 48 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 + 32t + 48 = 0.
There is t times t in it: a quadratic equation.
Now we solve it.
The t² part is negative.
We change every sign.
0 stays 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 − 2t − 3 = 0.
5 Factor
We need two numbers.
They multiply to −3, and they add to −2.
They multiply to a minus number, so one number is plus and one is minus.
They add to a minus number, so the bigger one is minus.
We list the pairs that make 3, smallest first.
We take the smaller from the bigger.
We need 2.
1 and 3 give 2.
Yes!
With the signs: 1 and −3.
So it is (t + 1)(t − 3) = 0.
6 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
We make each bracket 0.
Two small equations.
In t + 1 = 0, we take away 1 on both sides.
In t − 3 = 0, we add 3 on both sides.
So t = −1 or t = 3.
7 Answer the question
t = −1: before it is thrown: time is not negative.
So we throw it away.
Careful!
Both numbers make the equation true.
But a time before the throw makes no sense here.
Always look back at the story.
t = 3 works: it lands.
It lands after 3 seconds.
8 Check
Let's check in the story.
Put t = 3 into the height.
We get 0, on the ground.
So the answer is right!
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.