Difference between revisions of "Completing the Square"

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<math>(x-a)(x-a)=x^2-2ax+a^2</math><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>
 
The quadratic on the right is a perfect square trinomial. It is the square of a binomial.<br>
 +
====Example====
 
Take the example of  
 
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>
 
<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.
+
By completing the square, one of the components of the equation has to be a factored perfect square trinomial.<br>
 
+
====Relationship between terms====
 +
Note the relationship between the numbers in the factored form and the expanded form. The numbers available are 3, 6, and 9. The middle term of the quadratic is twice the constant of the binomial. The last term of the quadratic is the constant squared. In other words, <br>
 +
<math>2\times3=6 and 3\times3=9</math>
 
==Completing the Square==
 
==Completing the Square==
 
Let the quadratic equation be <math>ax^2+bx+c=0</math>
 
Let the quadratic equation be <math>ax^2+bx+c=0</math>
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<math>ax^2+bx=-c</math>
 
<math>ax^2+bx=-c</math>
 
*Step 2: Factor out the coefficient of the squared term <br>
 
*Step 2: Factor out the coefficient of the squared term <br>
<math>a(x^2+\tfrac{b}{a})=-c</math>
+
<math>a(x^2+\frac{b}{a})=-c</math>

Revision as of 13:41, 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.

Example

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.

Relationship between terms

Note the relationship between the numbers in the factored form and the expanded form. The numbers available are 3, 6, and 9. The middle term of the quadratic is twice the constant of the binomial. The last term of the quadratic is the constant squared. In other words,
2\times3=6 and 3\times3=9

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+\frac{b}{a})=-c