Solve x^2 - 3x = 4 by completing the square Step by Step

Step by step

The question

Question: solve x² − 3x = 4 by completing the square.

x² − 3x = 4

1 Complete the square

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

A picture shows what is missing.

So we add 9/4 to fill the corner.

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

x² − 3x + 94 = 4 + 94

2 Write it as a square

The full square is x − 3/2 on each side.

So the left side is (x − 3/2)².

x² − 3x + 94 = (x − 32)²

To add, both need the same bottom: 4.

4 is 16/4.

Same bottom, so we add the tops: 16 plus 9 is 25.

4 = 164

164 + 94 = 254

Careful!

Many people add 9/4 only on the left.

Then the two sides are not equal any more.

Careful: add 94 on BOTH sides

Now we have (x − 3/2)² = 25/4.

(x − 32)² = 254

3 Square root of both sides

Something squared is 25/4.

So that something is the square root of 25/4, 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.

5 times 5 is 25.

√254 = √25√4 = 52

So x − 3/2 is plus or minus that.

x − 32 = 52 or x − 32 = −52

4 Get x alone

To remove the − 3/2, we add 3/2 on both sides.

We do it in both equations.

x = 52 + 32 or x = −52 + 32

Same bottom, so we work with the tops.

5/2 plus 3/2 is 4.

Minus 5/2 plus 3/2 is −1.

52 + 32 = 82 = 4

−52 + 32 = −22 = −1

So x = 4 or x = −1.

x = 4 or x = −1

5 Check

Let's check: put x = 4 into the left side.

x = 4: 4² − 3(4) = 16 − 12 = 4

We get 4, the same as the right side.

It works!

Now put x = −1 into the left side.

x = −1: (−1)² − 3(−1) = 1 + 3 = 4

We get 4, 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