Difference between revisions of "Completing the Square"

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The equation<br>  
 
The equation<br>  
===<math> ax^2+bx+c</math><br>===
+
<math> ax^2+bx+c=0</math><br>
 
is converted into <br>
 
is converted into <br>
===<math>a(x-h)^2+k</math>===
+
<math>a(x-h)^2+k=0</math>
 
<br>
 
<br>
 
through the process of '''completing the square'''.  
 
through the process of '''completing the square'''.  
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==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
 ax^2+bx+c=0
is converted into
a(x-h)^2+k=0
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,
(x-a)(x-a)=x^2-2ax+a^2
The quadratic on the right is a perfect square trinomial. It is the square of a binomial.
Take the example of x^2+6x+9. Using basic algebra, it can be factored into (x+3)(x+3) or (x+3)^2
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 ax^2+bx+c=0

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

ax^2+bx=-c

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

a(x^2+\tfrac{b}{a})=-c