Worked examples
The question
Question: A ball is thrown up from the ground.
Its height after t seconds is h = -16t^2 + 64t feet.
When does it land?
1 Find what we know
What do we know?
The height formula: h = −16t² + 64t 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.
t = time in seconds
3 Write the equation
In words: the height is 0.
−16t2 + 64t = 0
So the equation is −16t² + 64t = 0.
There is t times t in it: a quadratic equation.
Now we solve it.
−16t2 + 64t = 0
The t² part is negative.
We change every sign.
0 stays 0.
16t2 − 64t = 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.
16, -64, 0: all divide by 16
Now we have t² − 4t = 0.
t2 − 4t = 0
5 Factor
There is no plain number.
Both parts have t in them.
So we take t out.
t² = t × t, −4t = t × (−4)
So it is t(t − 4) = 0.
t(t − 4) = 0
6 One bracket is 0
t times the bracket is 0.
That only happens if one of them is 0.
A × B = 0 → A = 0 or B = 0
We make each part 0.
Two small equations.
t = 0 or t − 4 = 0
t = 0 is done already.
In t − 4 = 0, we add 4 on both sides.
t = 0 or t − 4 + 4 = 0 + 4
So t = 0 or t = 4.
t = 0 or t = 4
7 Answer the question
t = 0: the moment it is thrown.
So we throw it away.
t = 0 ✗
the moment it is thrown
Careful!
At t = 0 the height is 0 too.
But that is when it is thrown, not when it lands.
Careful: t = 0 is the throw, not the landing
t = 4 works: it lands.
t = 4 ✓
it lands
It lands after 4 seconds.
8 Check
Let's check in the story.
Put t = 4 into the height.
We get 0, on the ground.
So the answer is right!
t = 4: h = −16(4)2 + 64(4) = −256 + 256 = 0
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.
▶ Watch it solved
The question
Question: A ball is thrown up from the ground.
Its height after t seconds is h = -16t^2 + 64t feet.
When is the ball 48 feet high?
1 Find what we know
What do we know?
The height formula: h = −16t² + 64t feet, after t seconds.
The question gives a height: 48 feet.
What do we want?
The times when the height is 48 feet.
2 Name the unknown
The formula already has a letter for the time: t, in seconds.
t = time in seconds
3 Write the equation
In words: the height is the number in the question.
−16t2 + 64t = 48
So the equation is −16t² + 64t = 48.
There is t times t in it: a quadratic equation.
Now we solve it.
−16t2 + 64t = 48
4 Get 0 on one side
To solve it, we want 0 on the right side.
So we take away 48 on both sides.
We do it on both sides, so they stay equal.
−16t2 + 64t − 48 = 48 − 48
Now we have −16t² + 64t − 48 = 0.
−16t2 + 64t − 48 = 0
The t² part is negative.
We change every sign on both sides.
0 stays 0.
16t2 − 64t + 48 = 0
5 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.
16, -64, 48: all divide by 16
Now we have t² − 4t + 3 = 0.
t2 − 4t + 3 = 0
6 Factor
We need two numbers.
They multiply to 3, and they add to −4.
? × ? = 3
? + ? = −4
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
× is + → same signs: both −
We list the pairs that make 3, smallest first.
We add each pair.
We need 4.
1 and 3 give 4.
Yes!
With the signs: −1 and −3.
1 × 3: 1 + 3 = 4 ✓
So it is (t − 1)(t − 3) = 0.
(t − 1)(t − 3) = 0
7 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
A × B = 0 → A = 0 or B = 0
We make each bracket 0.
Two small equations.
t − 1 = 0 or t − 3 = 0
In t − 1 = 0, we add 1 on both sides.
In t − 3 = 0, we add 3 on both sides.
t − 1 + 1 = 0 + 1 or t − 3 + 3 = 0 + 3
So t = 1 or t = 3.
t = 1 or t = 3
8 Answer the question
t = 1 works: going up.
t = 1 ✓
going up
t = 3 works: coming down.
t = 3 ✓
coming down
Careful!
Both times are answers.
It passes 48 feet going up, and again coming down.
Careful: not only t = 1 ✗
It is 48 feet high after 1 and after 3 seconds.
9 Check
Let's check in the story.
Put t = 1 into the height.
We get 48 feet.
t = 1: h = −16(1)2 + 64(1) = −16 + 64 = 48
Put t = 3 into the height.
We get 48 feet.
So the answer is right!
t = 3: h = −16(3)2 + 64(3) = −144 + 192 = 48
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.
▶ Watch it solved
The question
Question: A ball is thrown up from a roof.
Its height after t seconds is h = -16t^2 + 16t + 32 feet.
When does it hit the ground?
1 Find what we know
What do we know?
The height formula: h = −16t² + 16t + 32 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.
t = time in seconds
3 Write the equation
In words: the height is 0.
−16t2 + 16t + 32 = 0
So the equation is −16t² + 16t + 32 = 0.
There is t times t in it: a quadratic equation.
Now we solve it.
−16t2 + 16t + 32 = 0
The t² part is negative.
We change every sign.
0 stays 0.
16t2 − 16t − 32 = 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.
16, -16, -32: all divide by 16
Now we have t² − t − 2 = 0.
t2 − t − 2 = 0
5 Factor
We need two numbers.
They multiply to −2, and they add to −1.
? × ? = −2
? + ? = −1
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.
× is − → one +, one −; bigger one −
We list the pairs that make 2, smallest first.
We take the smaller from the bigger.
We need 1.
1 and 2 give 1.
Yes!
With the signs: 1 and −2.
1 × 2: 2 − 1 = 1 ✓
So it is (t + 1)(t − 2) = 0.
(t + 1)(t − 2) = 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.
A × B = 0 → A = 0 or B = 0
We make each bracket 0.
Two small equations.
t + 1 = 0 or t − 2 = 0
In t + 1 = 0, we take away 1 on both sides.
In t − 2 = 0, we add 2 on both sides.
t + 1 − 1 = 0 − 1 or t − 2 + 2 = 0 + 2
So t = −1 or t = 2.
t = −1 or t = 2
7 Answer the question
t = −1: before it is thrown: time is not negative.
So we throw it away.
t = −1 ✗
before it is thrown: time is not negative
Careful!
Both numbers make the equation true.
But a time before the throw makes no sense here.
Always look back at the story.
Careful: t = −1 fits the equation, not the story
t = 2 works: it lands.
t = 2 ✓
it lands
It lands after 2 seconds.
8 Check
Let's check in the story.
Put t = 2 into the height.
We get 0, on the ground.
So the answer is right!
t = 2: h = −16(2)2 + 16(2) + 32 = −64 + 32 + 32 = 0
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.
▶ Watch it solved
