Solve x^2 - 2x - 7 = 0 by completing the square Step by Step

Step by step

The question

Question: solve x² − 2x − 7 = 0 by completing the square.

x² − 2x − 7 = 0

1 Move the −7

We want only the x parts on the left.

The − 7 is in the way.

So we add 7 on both sides.

x² − 2x ~~− 7 + 7~~ = 0 + 7

− 7 and + 7 make zero.

Cross them out.

0 plus 7 is 7.

0 + 7 = 7

Now we have x² − 2x = 7.

x² − 2x = 7

2 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 = 7 + 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

3 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

7 plus 1 is 8.

7 + 1 = 8

Now we have (x − 1)² = 8.

(x − 1)² = 8

4 Square root of both sides

Something squared is 8.

So that something is the square root of 8, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

8 is not a square number, so we keep √8.

x − 1 = √8 or x − 1 = −√8

5 Get x alone

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

We do it in both equations.

x = √8 + 1 or x = −√8 + 1

We write the plain number first.

So x = 1 + √8 or x = 1 − √8.

x = 1 + √8 or x = 1 − √8

With a calculator, √8 is about 2.83.

So x is about 3.83 or about −1.83.

x ≈ 3.83 or x ≈ −1.83

Let's look back.

First: move the −7.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Last: get x alone.

Answer