How many solutions 4x^2 + 12x + 9 = 0 Step by Step

Step by step

The question

Question: find the number of real solutions of 4x² + 12x + 9 = 0.

4x² + 12x + 9 = 0

1 Find a, b and c

We read the numbers: a = 4 is with x², b = 12 is with x, c = 9 is the plain number.

a = 4, b = 12, 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 = 12² − 4 × 4 × 9

12² is 144.

4 times 4 times 9 is 144.

12² = 12 × 12 = 144

4 × 4 × 9 = 144

Careful!

b is squared first: b² means 12 times 12, not 12.

Careful: not 12 − 144 ✗

So we have 144 take away 144.

= 144 − 144

So the discriminant is 0.

= 0

3 How many solutions

It is 0.

The square root of 0 is 0.

Plus 0 and minus 0 give the same answer.

So there is just one.

positive → 2 real solutions

0 → 1 real solution

negative → no real solutions

So 4x² + 12x + 9 = 0 has 1 real solution.

= 1 real solution

4 Check

Let's check with the formula.

Under the root we get 0, and the square root of 0 is 0.

x = −12 ± √08 = −12 ± 08

Plus 0 and minus 0 give the same number: x = −3/2.

One solution.

It works!

x = −32 only

Let's look back.

First: find a, b and c.

Then: work out b² − 4ac.

Then: how many solutions.

Last: check.

Answer