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

Step by step

The question

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

x² + x + 1 = 0

1 Find a, b and c

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

a = 1, b = 1, 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 = 1² − 4 × 1 × 1

1² is 1.

4 times 1 times 1 is 4.

1² = 1 × 1 = 1

4 × 1 × 1 = 4

So we have 1 take away 4.

= 1 − 4

So the discriminant is −3.

= −3

3 How many solutions

−3 is negative.

No real number times itself is negative.

So the square root of −3 is not a real number.

positive → 2 real solutions

0 → 1 real solution

negative → no real solutions

So x² + x + 1 = 0 has no real solutions.

= no real solutions

4 Check

Let's check with the formula.

Under the root we get −3.

x = −1 ± √−32

No real number squared is negative.

So the formula gives no real solutions.

It works!

√(−3): 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