Step by step
The question
Question: find the number of real solutions of 2x² − 7x + 3 = 0.
1 Find a, b and c
We read the numbers: a = 2 is with x², b = −7 is with x, c = 3 is the plain number.
2 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.
(−7)² is 49.
Minus times minus is plus.
4 times 2 times 3 is 24.
Careful!
b = −7 goes in brackets.
(−7) squared is 49, not minus 49.
So we have 49 take away 24.
So the discriminant is 25.
3 How many solutions
25 is positive.
The square root of a positive number is a real number.
The plus or minus gives two different answers.
So 2x² − 7x + 3 = 0 has 2 real solutions.
4 Check
Let's check with the formula.
Under the root we get 25.
Its square root is 5.
The plus gives 3, the minus gives 1/2.
Two different solutions.
It works!
Let's look back.
First: find a, b and c.
Then: work out b² − 4ac.
Then: how many solutions.
Last: check.