Solve x^2 - 5x = 14 by completing the square Step by Step

Step by step

The question

Question: solve x² − 5x = 14 by completing the square.

x² − 5x = 14

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 = 14 + 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.

14 is 56/4.

Same bottom, so we add the tops: 56 plus 25 is 81.

14 = 564

564 + 254 = 814

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 = 81/4.

(x − 52)² = 814

3 Square root of both sides

Something squared is 81/4.

So that something is the square root of 81/4, 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.

9 times 9 is 81.

√814 = √81√4 = 92

So x − 5/2 is plus or minus that.

x − 52 = 92 or x − 52 = −92

4 Get x alone

To remove the − 5/2, we add 5/2 on both sides.

We do it in both equations.

x = 92 + 52 or x = −92 + 52

Same bottom, so we work with the tops.

9/2 plus 5/2 is 7.

Minus 9/2 plus 5/2 is −2.

92 + 52 = 142 = 7

−92 + 52 = −42 = −2

So x = 7 or x = −2.

x = 7 or x = −2

5 Check

Let's check: put x = 7 into the left side.

x = 7: 7² − 5(7) = 49 − 35 = 14

We get 14, the same as the right side.

It works!

Now put x = −2 into the left side.

x = −2: (−2)² − 5(−2) = 4 + 10 = 14

We get 14, 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