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# Completing the Square

(KS4, Year 10)

## The Lesson

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**and

^{2}**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}+ 4xSplit 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**in the space... we will

^{2}**complete the square**:

Each side of this larger square is

**x + 2**. Its area is

**(x + 2)**. By adding up the areas we see that:

^{2}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} |

**x**not

^{2}+ 4x**x**. We added

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

^{2}**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:**Help Us To Improve Mathematics Monster**

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