Solve x^2 + 3x = 10 by completing the square Step by Step

Step by step

The question

Question: solve x² + 3x = 10 by completing the square.

x² + 3x = 10

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 = 10 + 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.

10 is 40/4.

Same bottom, so we add the tops: 40 plus 9 is 49.

10 = 404

404 + 94 = 494

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 = 49/4.

(x + 32)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 + 3/2 is plus or minus that.

x + 32 = 72 or x + 32 = −72

4 Get x alone

To remove the + 3/2, we take away 3/2 on both sides.

We do it in both equations.

x = 72 − 32 or x = −72 − 32

Same bottom, so we work with the tops.

7/2 minus 3/2 is 2.

Minus 7/2 minus 3/2 is −5.

72 − 32 = 42 = 2

−72 − 32 = −102 = −5

So x = 2 or x = −5.

x = 2 or x = −5

5 Check

Let's check: put x = 2 into the left side.

x = 2: 2² + 3(2) = 4 + 6 = 10

We get 10, the same as the right side.

It works!

Now put x = −5 into the left side.

x = −5: (−5)² + 3(−5) = 25 − 15 = 10

We get 10, 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