Step by step
The question
Question: find the number of real solutions of 2x² = 3x + 5.
1 Get 0 on one side
We want 0 on the right side.
So we take away 3x + 5 on both sides.
We do it on both sides, so they stay equal.
Now we have 2x² − 3x − 5 = 0.
2 Find a, b and c
We read the numbers: a = 2 is with x², b = −3 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.
(−3)² is 9.
Minus times minus is plus.
4 times 2 times −5 is −40.
Careful!
b = −3 goes in brackets.
(−3) squared is 9, not minus 9.
So we have 9 take away −40.
Taking away a negative is the same as adding.
So the discriminant is 49.
4 How many solutions
49 is positive.
The square root of a positive number is a real number.
The plus or minus gives two different answers.
So 2x² = 3x + 5 has 2 real solutions.
5 Check
Let's check with the formula.
Under the root we get 49.
Its square root is 7.
The plus gives 5/2, the minus gives −1.
Two different 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.