Step by step
The question
Question: find the number of real solutions of x² = 6x − 9.
1 Get 0 on one side
We want 0 on the right side.
So we take away 6x − 9 on both sides.
We do it on both sides, so they stay equal.
Now we have x² − 6x + 9 = 0.
2 Find a, b and c
We read the numbers: a = 1 is with x², b = −6 is with x, c = 9 is the plain number.
3 Work out b² − 4ac
In the formula, b² − 4ac is under the square root.
It is called the discriminant.
Its sign tells us how many solutions there are.
We put the numbers in, each negative one in brackets.
(−6)² is 36.
Minus times minus is plus.
4 times 1 times 9 is 36.
Careful!
b = −6 goes in brackets.
(−6) squared is 36, not minus 36.
So we have 36 take away 36.
So the discriminant is 0.
4 How many solutions
It is 0.
The square root of 0 is 0.
Plus 0 and minus 0 give the same answer.
So there is just one.
So x² = 6x − 9 has 1 real solution.
5 Check
Let's check with the formula.
Under the root we get 0, and the square root of 0 is 0.
Plus 0 and minus 0 give the same number: x = 3.
One solution.
It works!
Let's look back.
First: get 0 on one side.
Then: find a, b and c.
Then: work out b² − 4ac.
Then: how many solutions.
Last: check.