# What Is Completing the Square?

## Completing the Square

Completing the square is a way of simplifying a quadratic equation.

From perfect square trinomials, we have seen that a squared binomial expands to a quadratic equation:

If we have the first two terms in a quadratic equation (the **x ^{2}** and

**x**terms), we can write it as a squared binomial minus a number:

## Understanding Completing the Square Using Equations

Let's look at the patterns in the equations:

### Halve the Number in Front of the x...

The number in the brackets (**2**) is **half** the number in front of the **x** (**4**).

### ...Then Square and Subtract It

The number being subtracted from the squared brackets (**2**) is **half** the number in front of the **x** (**4**) squared (**2 ^{2}**).

## Understanding Completing the Square Using Geometry

The area of the square plus the area of the rectangle below equals **x ^{2} + 4x**:

Split the rectangle in half. Instead of one rectangle with an area of **4x**, there are two rectangles each with an area of **2x**:

Place the two rectangles by the side of the square:

We almost have a square. If we place a small square of area **2 ^{2}** in the space... we will

**complete the square**:

Each side of this larger square is **x + 2**. Its area is **(x + 2) ^{2}**.

By adding up the areas we see that:

x^{2} + 2x + 2x + 2^{2} = (x + 2)^{2} |
4 small areas = 1 big area |

x^{2} + 2x + 2x + 2^{2} = (x + 2)^{2} |
Add like terms |

x^{2} + 4x + 2^{2} = (x + 2)^{2} |

The original area was **x ^{2} + 4x** not

**x**.

^{2}+ 4x + 2^{2}We added **2 ^{2}** to complete the square. To balance the equation, we must subtract

**2**from both sides:

^{2}x^{2} + 4x + 2^{2} − 2 = (x + 2)^{2}^{2} − 2^{2} |
Subtract 2 from both sides^{2} |

x^{2} + 4x = (x + 2)^{2} − 2^{2} |

## Completing the Square in General

In general, we complete the square as: