How many solutions x^2 - 7 = 0 Step by Step

Step by step

The question

Question: find the number of real solutions of x² − 7 = 0.

x² − 7 = 0

1 Find a, b and c

We read the numbers: a = 1 is with x², b = 0 is with x, c = −7 is the plain number.

There is no x term, so b is 0.

a = 1, b = 0, c = −7

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 = 0² − 4 × 1 × (−7)

0² is 0.

4 times 1 times −7 is −28.

0² = 0 × 0 = 0

4 × 1 × (−7) = −28

Careful!

c is negative, so 4ac is negative.

Taking away −28 is adding 28.

Careful: not 0 − 28 ✗

So we have 0 take away −28.

= 0 − (−28)

Taking away a negative is the same as adding.

0 − (−28) = 0 + 28

So the discriminant is 28.

= 28

3 How many solutions

28 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 − 7 = 0 has 2 real solutions.

= 2 real solutions

4 Check

Let's check with the formula.

Under the root we get 28.

x = 0 ± √282

With a calculator: about 2.6 and about −2.6.

Two different solutions.

It works!

x ≈ 2.6 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