Step by step
The question
Question: find the number of real solutions of x² + 5 = 2x.
1 Get 0 on one side
We want 0 on the right side.
So we take away 2x on both sides.
We do it on both sides, so they stay equal.
Now we have x² − 2x + 5 = 0.
2 Find a, b and c
We read the numbers: a = 1 is with x², b = −2 is with x, c = 5 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.
(−2)² is 4.
Minus times minus is plus.
4 times 1 times 5 is 20.
Careful!
b = −2 goes in brackets.
(−2) squared is 4, not minus 4.
So we have 4 take away 20.
So the discriminant is −16.
4 How many solutions
−16 is negative.
No real number times itself is negative.
So the square root of −16 is not a real number.
So x² + 5 = 2x has no real solutions.
5 Check
Let's check with the formula.
Under the root we get −16.
No real number squared is negative.
So the formula gives no real solutions.
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.