Completing the Square

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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