Solve x^2 + 5x = 6 by completing the square Step by Step

Step by step

The question

Question: solve x² + 5x = 6 by completing the square.

x² + 5x = 6

1 Complete the square

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

A picture shows what is missing.

So we add 25/4 to fill the corner.

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

x² + 5x + 254 = 6 + 254

2 Write it as a square

The full square is x + 5/2 on each side.

So the left side is (x + 5/2)².

x² + 5x + 254 = (x + 52)²

To add, both need the same bottom: 4.

6 is 24/4.

Same bottom, so we add the tops: 24 plus 25 is 49.

6 = 244

244 + 254 = 494

Careful!

Many people add 25/4 only on the left.

Then the two sides are not equal any more.

Careful: add 254 on BOTH sides

Now we have (x + 5/2)² = 49/4.

(x + 52)² = 494

3 Square root of both sides

Something squared is 49/4.

So that something is the square root of 49/4, or minus it.

Both work, because minus times minus is plus.

( )² = k → ( ) = √k or −√k

The square root of a fraction is the root of the top over the root of the bottom.

2 times 2 is 4.

7 times 7 is 49.

√494 = √49√4 = 72

So x + 5/2 is plus or minus that.

x + 52 = 72 or x + 52 = −72

4 Get x alone

To remove the + 5/2, we take away 5/2 on both sides.

We do it in both equations.

x = 72 − 52 or x = −72 − 52

Same bottom, so we work with the tops.

7/2 minus 5/2 is 1.

Minus 7/2 minus 5/2 is −6.

72 − 52 = 22 = 1

−72 − 52 = −122 = −6

So x = 1 or x = −6.

x = 1 or x = −6

5 Check

Let's check: put x = 1 into the left side.

x = 1: 1² + 5(1) = 1 + 5 = 6

We get 6, the same as the right side.

It works!

Now put x = −6 into the left side.

x = −6: (−6)² + 5(−6) = 36 − 30 = 6

We get 6, 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