Completing the Square

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The Basics

Completing the Square is a method used to solve quadratic equations. When a quadratic is hard to factor or not factorable at all, this method rewrites the quadratic equation originally in standard form into vertex form which like a factored quadratic is much easier to solve.

The equation
is converted into

and x=\pm\sqrt{\frac{-k}{a}}+h

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 left is a perfect square trinomial. It is the square of a binomial.

Example of a Perfect Square Trinomial

Take the example of x^2+6x+9. It can be factored into (x+3)(x+3) whcih is equal to (x+3)^2
This concept is important because in the process of completing the square, one of the components of the equation has to be a factored perfect square trinomial.

Relationship between terms

  • Note that the coefficient of the middle term is twice the square root of the constant term in a quadratic equation.
  • In other words, the constant term is the square of half of the coefficient of x.
  • NOTE that this holds only true if the coefficient of x^2 is 1.

In general given the perfect trinomial and the equivalent squared binomial x^2+bx+c=(x+d)^2
Then x^2+bx+c=x^2+2dx+d^2
b=2d and c=d^2
We can rearrange equalities to obtain the following: c=(\tfrac{b}{2})^2

Procedure for Completing the Square

Let the quadratic equation be ax^2+bx+c=0

  • Step 1 : Move the constant term c over to the other side of the equal sign.
  • Step 2 : Factor out a, the coefficient of the squared term.
  • Step 3 : Complete the quadratic in the parenthesis to make a perfect square trinomial. Remember that in general the constant of the perfect square trinomial is the square of half of the coefficient of x.
  • Step 4 : Add the equivalent added to the left side of the equation, to the right side of the equation to maintain the equality. Remember to multiply the obtained constant by a when added to the right side.
  • Step 5 : Factor the left side, simplify the right side and bring it over to the left.
  • Step 6(optional): If asked for, solve for x.


Though the process of completing the square is finalized, it does not look like the vertex form equation

Let <template>AlignEquals


We can now substitute these values into the equation in Step 5 and obtain

Example with a = 1

All the different letters are bound to get confusing. It may be easier to understand the process using an example with real numbers. Suppose we are given the quadratic x^2-12x+5, and asked to solve for x. The first thing we do is see if it can be factored: it can't. So to solve for x, we will complete the square.

Example with a not 1

Given 3x^2+2x-7