# Difference between revisions of "Completing the Square"

Line 3: | Line 3: | ||

The equation<br> | The equation<br> | ||

− | + | <math> ax^2+bx+c=0</math><br> | |

is converted into <br> | is converted into <br> | ||

− | + | <math>a(x-h)^2+k=0</math> | |

<br> | <br> | ||

through the process of '''completing the square'''. | through the process of '''completing the square'''. | ||

Line 11: | Line 11: | ||

==Perfect Square Trinomial== | ==Perfect Square Trinomial== | ||

A '''perfect square trinomial''' is a big fancy word for a simple concept. It is a quadratic equation that factors perfectly into two identical binomials.<br> | A '''perfect square trinomial''' is a big fancy word for a simple concept. It is a quadratic equation that factors perfectly into two identical binomials.<br> | ||

− | In general | + | In general, <br> |

+ | <math>(x-a)(x-a)=x^2-2ax+a^2</math><br> | ||

+ | The quadratic on the right is a perfect square trinomial. It is the square of a binomial.<br> | ||

+ | Take the example of | ||

+ | <math>x^2+6x+9</math>. Using basic algebra, it can be factored into <math>(x+3)(x+3)</math> or <math>(x+3)^2</math><br> | ||

+ | By completing the square, one of the components of the equation has to be a factored perfect square trinomial. | ||

+ | |||

+ | ==Completing the Square== | ||

+ | Let the quadratic equation be <math>ax^2+bx+c=0</math> | ||

+ | *Step 1: Move the constant over to the other side of the equality | ||

+ | <math>ax^2+bx=-c</math> | ||

+ | *Step 2: Factor out the coefficient of the squared term <br> | ||

+ | <math>a(x^2+\tfrac{b}{a})=-c</math> |

## Revision as of 12:57, 15 June 2009

## The Basics

**Completing the Square** is a method commonly used to solve quadratic equations. Often times, a quadratic equation can be factored and solved easily. However, there are plenty of times when an equation is not factorable. By completing the square, a quadratic equation originally in standard form is rewritten into vertex form.

The equation

is converted into

through the process of **completing the square**.

## Perfect Square Trinomial

A **perfect square trinomial** is a big fancy word for a simple concept. It is a quadratic equation that factors perfectly into two identical binomials.

In general,

The quadratic on the right is a perfect square trinomial. It is the square of a binomial.

Take the example of
. Using basic algebra, it can be factored into or

By completing the square, one of the components of the equation has to be a factored perfect square trinomial.

## Completing the Square

Let the quadratic equation be

- Step 1: Move the constant over to the other side of the equality

- Step 2: Factor out the coefficient of the squared term