Step by step
The question
Question: find the number of real solutions of 2x² + 9x − 5 = 0.
1 Find a, b and c
We read the numbers: a = 2 is with x², b = 9 is with x, c = −5 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.
9² is 81.
4 times 2 times −5 is −40.
Careful!
c is negative, so 4ac is negative.
Taking away −40 is adding 40.
So we have 81 take away −40.
Taking away a negative is the same as adding.
So the discriminant is 121.
3 How many solutions
121 is positive.
The square root of a positive number is a real number.
The plus or minus gives two different answers.
So 2x² + 9x − 5 = 0 has 2 real solutions.
4 Check
Let's check with the formula.
Under the root we get 121.
Its square root is 11.
The plus gives 1/2, the minus gives −5.
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.