# Algebra/Completing the Square

## Derivation[edit]

The purpose of "completing the square" is to either factor a prime quadratic equation or to more easily graph a parabola. The procedure to follow is as follows for a quadratic equation :

1. Divide everything by *a*, so that the number in front of is a perfect square (1):

2. Now we want to focus on the term in front of the *x*. Add the quantity to both sides:

3. Now notice that on the right, the first three terms factor into a perfect square:

Multiply this back out to convince yourself that this works.

4. Therefore the *completed square* form of the quadratic is:

- or, multiplying through by a,

## Explanation of Derivation[edit]

1. Divide everything by *a*, so that the number in front of is a perfect square (1):

Think of this as expressing your final result in terms of 1 square x. If your initial equation is

2. Now we want to focus on the term in front of the *x*. Add the quantity to both sides:

3. Now notice that on the right, the first three terms factor into a perfect square:

Multiply this back out to convince yourself that this works.

4. Therefore the *completed square* form of the quadratic is:

- or, multiplying through by a,

## Example[edit]

The best way to learn to complete a square is through an example. Suppose you want to solve the following equation for x.

2x^{2} + 24x + 23 = 0 |
Does not factor easily, so we complete the square. |

x^{2} + 12x + 23/2 = 0 |
Make coefficient of x^{2} a 1, by dividing all terms by 2. |

x^{2} + 12x = - 23/2 |
Add – 23/2 to both sides. |

x^{2} + 12x + 36 = - 23/2 + 36 |
Take half of 12 (coefficient of x), and square it. Add to both sides. |

(x + 6)^{2} = 49/2 |
Factor. Now we can take square roots to easily solve this form of the equation. |

√(x + 6)^{2} = √49/√2 |
Take the square root. |

x + 6 = 7/√2 | Simplify. |

x = -6 + (7√2)/2 | Rationalize the denominator. |