Step by step
The question
Question: find x when f(x) = 8 for f(x) = x² − 2x.
f(x) = 8 means: the answer that comes out is 8.
We look for the x that goes in.
Careful!
It is not f(8).
8 is what comes out, not what goes in.
1 Write the equation
f(x) is x² − 2x.
So x² − 2x = 8.
Now we solve it.
2 Get 0 on one side
To solve it, we want 0 on the right side.
So we take away 8 on both sides.
We do it on both sides, so they stay equal.
Now we have x² − 2x − 8 = 0.
3 Factor
We need two numbers.
They multiply to −8, 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 8, smallest first.
We take the smaller from the bigger.
We need 2.
2 and 4 give 2.
Yes!
With the signs: 2 and −4.
So it is (x + 2)(x − 4) = 0.
4 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 x + 2 = 0, we take away 2 on both sides.
In x − 4 = 0, we add 4 on both sides.
So x = −2 or x = 4.
5 Check
Let's check: put −2 into f.
We get 8.
And 4 into f.
We get 8 too.
So f(−2) = 8 and f(4) = 8.
Both answers are right!
Let's look back.
First: write the equation.
Then: get 0 on one side.
Then: factor.
Then: one bracket is 0.
Last: check.