How many solutions 3x^2 + 2x + 9 = 0 Step by Step

Step by step

The question

Question: find the number of real solutions of 3x² + 2x + 9 = 0.

3x² + 2x + 9 = 0

1 Find a, b and c

We read the numbers: a = 3 is with x², b = 2 is with x, c = 9 is the plain number.

a = 3, b = 2, c = 9

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.

x = −b ± √b² − 4ac2a

We put the numbers in, each negative one in brackets.

b² − 4ac = 2² − 4 × 3 × 9

2² is 4.

4 times 3 times 9 is 108.

2² = 2 × 2 = 4

4 × 3 × 9 = 108

Careful!

b is squared first: b² means 2 times 2, not 2.

Careful: not 2 − 108 ✗

So we have 4 take away 108.

= 4 − 108

So the discriminant is −104.

= −104

3 How many solutions

−104 is negative.

No real number times itself is negative.

So the square root of −104 is not a real number.

positive → 2 real solutions

0 → 1 real solution

negative → no real solutions

So 3x² + 2x + 9 = 0 has no real solutions.

= no real solutions

4 Check

Let's check with the formula.

Under the root we get −104.

x = −2 ± √−1046

No real number squared is negative.

So the formula gives no real solutions.

It works!

√(−104): not a real number

Let's look back.

First: find a, b and c.

Then: work out b² − 4ac.

Then: how many solutions.

Last: check.

Answer