How many solutions x^2 + 3x + 1 = 0 Step by Step

Step by step

The question

Question: find the number of real solutions of x² + 3x + 1 = 0.

x² + 3x + 1 = 0

1 Find a, b and c

We read the numbers: a = 1 is with x², b = 3 is with x, c = 1 is the plain number.

a = 1, b = 3, c = 1

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 = 3² − 4 × 1 × 1

3² is 9.

4 times 1 times 1 is 4.

3² = 3 × 3 = 9

4 × 1 × 1 = 4

Careful!

b is squared first: b² means 3 times 3, not 3.

Careful: not 3 − 4 ✗

So we have 9 take away 4.

= 9 − 4

So the discriminant is 5.

= 5

3 How many solutions

5 is positive.

The square root of a positive number is a real number.

The plus or minus gives two different answers.

positive → 2 real solutions

0 → 1 real solution

negative → no real solutions

So x² + 3x + 1 = 0 has 2 real solutions.

= 2 real solutions

4 Check

Let's check with the formula.

Under the root we get 5.

x = −3 ± √52

With a calculator: about −0.4 and about −2.6.

Two different solutions.

It works!

x ≈ −0.4 or x ≈ −2.6

Let's look back.

First: find a, b and c.

Then: work out b² − 4ac.

Then: how many solutions.

Last: check.

Answer