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

Step by step

The question

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

x² + 2x − 2 = 0

1 Find a, b and c

We read the numbers: a = 1 is with x², b = 2 is with x, c = −2 is the plain number.

a = 1, b = 2, c = −2

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 × 1 × (−2)

2² is 4.

4 times 1 times −2 is −8.

2² = 2 × 2 = 4

4 × 1 × (−2) = −8

Careful!

c is negative, so 4ac is negative.

Taking away −8 is adding 8.

Careful: not 4 − 8 ✗

So we have 4 take away −8.

= 4 − (−8)

Taking away a negative is the same as adding.

4 − (−8) = 4 + 8

So the discriminant is 12.

= 12

3 How many solutions

12 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 + 2x − 2 = 0 has 2 real solutions.

= 2 real solutions

4 Check

Let's check with the formula.

Under the root we get 12.

x = −2 ± √122

With a calculator: about 0.7 and about −2.7.

Two different solutions.

It works!

x ≈ 0.7 or x ≈ −2.7

Let's look back.

First: find a, b and c.

Then: work out b² − 4ac.

Then: how many solutions.

Last: check.

Answer