Solve x^2 - 6x + 4 = 0 by completing the square Step by Step

Step by step

The question

Question: solve x² − 6x + 4 = 0 by completing the square.

x² − 6x + 4 = 0

1 Move the 4

We want only the x parts on the left.

The + 4 is in the way.

So we take away 4 on both sides.

x² − 6x ~~+ 4 − 4~~ = 0 − 4

+ 4 and − 4 make zero.

Cross them out.

0 minus 4 is −4.

0 − 4 = −4

Now we have x² − 6x = −4.

x² − 6x = −4

2 Complete the square

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

A picture shows what is missing.

So we add 9 to fill the corner.

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

x² − 6x + 9 = −4 + 9

Careful!

Many people add 9 only on the left.

Then the two sides are not equal any more.

Careful: add 9 on BOTH sides

3 Write it as a square

The full square is x − 3 on each side.

So x² − 6x + 9 is (x − 3)².

x² − 6x + 9 = (x − 3)²

−4 plus 9 is 5.

−4 + 9 = 5

Now we have (x − 3)² = 5.

(x − 3)² = 5

4 Square root of both sides

Something squared is 5.

So that something is the square root of 5, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

5 is not a square number, so we keep √5.

x − 3 = √5 or x − 3 = −√5

5 Get x alone

To remove the − 3, we add 3 on both sides.

We do it in both equations.

x = √5 + 3 or x = −√5 + 3

We write the plain number first.

So x = 3 + √5 or x = 3 − √5.

x = 3 + √5 or x = 3 − √5

With a calculator, √5 is about 2.24.

So x is about 5.24 or about 0.76.

x ≈ 5.24 or x ≈ 0.76

Let's look back.

First: move the 4.

Then: complete the square.

Then: write it as a square.

Then: square root of both sides.

Last: get x alone.

Answer