Solve x^2 + 2x + 5 = 0 by completing the square Step by Step

Step by step

The question

Question: solve x² + 2x + 5 = 0 by completing the square.

x² + 2x + 5 = 0

1 Move the 5

We want only the x parts on the left.

The + 5 is in the way.

So we take away 5 on both sides.

x² + 2x ~~+ 5 − 5~~ = 0 − 5

+ 5 and − 5 make zero.

Cross them out.

0 minus 5 is −5.

0 − 5 = −5

Now we have x² + 2x = −5.

x² + 2x = −5

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 = −5 + 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

−5 plus 1 is −4.

−5 + 1 = −4

Now we have (x + 1)² = −4.

(x + 1)² = −4

4 Square root of both sides

Something squared is −4.

( )² = k → ( ) = √k or −√k

But no real number squared is negative.

So there is no real solution.

x + 1 = no real solution

Let's look back.

First: move the 5.

Then: complete the square.

Then: write it as a square.

Last: square root of both sides.

Answer