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

Step by step

The question

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

x² − 2x = 0

1 Find a, b and c

We read the numbers: a = 1 is with x², b = −2 is with x, c = 0 is the plain number.

There is no plain number, so c is 0.

a = 1, b = −2, c = 0

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 = (−2)² − 4 × 1 × 0

(−2)² is 4.

Minus times minus is plus.

4 times 1 times 0 is 0.

(−2)² = (−2) × (−2) = 4

4 × 1 × 0 = 0

Careful!

b = −2 goes in brackets.

(−2) squared is 4, not minus 4.

Careful: not −2² = −4 ✗

So we have 4 take away 0.

= 4 − 0

So the discriminant is 4.

= 4

3 How many solutions

4 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 = 0 has 2 real solutions.

= 2 real solutions

4 Check

Let's check with the formula.

Under the root we get 4.

Its square root is 2.

x = 2 ± 22

The plus gives 2, the minus gives 0.

Two different solutions.

It works!

x = 2 or 0

Let's look back.

First: find a, b and c.

Then: work out b² − 4ac.

Then: how many solutions.

Last: check.

Answer