Step by step
The question
Question: A stone is thrown up.
Its height after t seconds is h = -5t^2 + 20t meters.
When does it land?
1 Find what we know
What do we know?
The height formula: h = −5t² + 20t meters, 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 −5t² + 20t = 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 5
Easy way first: every number can be divided by 5.
So we divide both sides by 5.
0 divided by 5 is still 0.
Now we have t² − 4t = 0.
5 Factor
There is no plain number.
Both parts have t in them.
So we take t out.
So it is t(t − 4) = 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 − 4 = 0, we add 4 on both sides.
So t = 0 or t = 4.
7 Answer the question
t = 0: the moment it is thrown.
So we throw it away.
Careful!
At t = 0 the height is 0 too.
But that is when it is thrown, not when it lands.
t = 4 works: 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!
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.