Solve x^2 - 2x = 3 by completing the square Step by Step

Step by step

The question

Question: solve x² − 2x = 3 by completing the square.

x² − 2x = 3

1 Complete the square

We want the left side to be one bracket, squared.

A picture shows what is missing.

So we add 1 to fill the corner.

We do it on both sides, so they stay equal.

x² − 2x + 1 = 3 + 1

Careful!

Many people add 1 only on the left.

Then the two sides are not equal any more.

Careful: add 1 on BOTH sides

2 Write it as a square

The full square is x − 1 on each side.

So x² − 2x + 1 is (x − 1)².

x² − 2x + 1 = (x − 1)²

3 plus 1 is 4.

3 + 1 = 4

Now we have (x − 1)² = 4.

(x − 1)² = 4

3 Square root of both sides

Something squared is 4.

So that something is the square root of 4, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

The square root of 4 is 2, because 2 times 2 is 4.

x − 1 = 2 or x − 1 = −2

4 Get x alone

To remove the − 1, we add 1 on both sides.

We do it in both equations.

x = 2 + 1 or x = −2 + 1

2 plus 1 is 3.

Minus 2 plus 1 is −1.

2 + 1 = 3, −2 + 1 = −1

So x = 3 or x = −1.

x = 3 or x = −1

5 Check

Let's check: put x = 3 into the left side.

x = 3: 3² − 2(3) = 9 − 6 = 3

We get 3, the same as the right side.

It works!

Now put x = −1 into the left side.

x = −1: (−1)² − 2(−1) = 1 + 2 = 3

We get 3, the same as the right side.

So both answers are right!

Let's look back.

First: complete the square.

Then: write it as a square.

Then: square root of both sides.

Then: get x alone.

Last: check.

Answer