Solve x^2 - 8x + 13 = 0 by completing the square Step by Step

Step by step

The question

Question: solve x² − 8x + 13 = 0 by completing the square.

x² − 8x + 13 = 0

1 Move the 13

We want only the x parts on the left.

The + 13 is in the way.

So we take away 13 on both sides.

x² − 8x ~~+ 13 − 13~~ = 0 − 13

+ 13 and − 13 make zero.

Cross them out.

0 minus 13 is −13.

0 − 13 = −13

Now we have x² − 8x = −13.

x² − 8x = −13

2 Complete the square

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

A picture shows what is missing.

So we add 16 to fill the corner.

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

x² − 8x + 16 = −13 + 16

Careful!

Many people add 16 only on the left.

Then the two sides are not equal any more.

Careful: add 16 on BOTH sides

3 Write it as a square

The full square is x − 4 on each side.

So x² − 8x + 16 is (x − 4)².

x² − 8x + 16 = (x − 4)²

−13 plus 16 is 3.

−13 + 16 = 3

Now we have (x − 4)² = 3.

(x − 4)² = 3

4 Square root of both sides

Something squared is 3.

So that something is the square root of 3, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

3 is not a square number, so we keep √3.

x − 4 = √3 or x − 4 = −√3

5 Get x alone

To remove the − 4, we add 4 on both sides.

We do it in both equations.

x = √3 + 4 or x = −√3 + 4

We write the plain number first.

So x = 4 + √3 or x = 4 − √3.

x = 4 + √3 or x = 4 − √3

With a calculator, √3 is about 1.73.

So x is about 5.73 or about 2.27.

x ≈ 5.73 or x ≈ 2.27

Let's look back.

First: move the 13.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Last: get x alone.

Answer