Completing the Square

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The Basics

Completing the Square is a method used to solve quadratic equations. In fact, any quadratic equation can be solved using this method. When a quadratic is difficult to factor or it involves complex numbers, 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 quadratic equation that factors perfectly into two identical binomials.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.
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. The non-factored expression is the 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 by adding the square of half of the coefficient. Add the equivalent value 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 4 : Factor the left side, simplify the right side and bring it over to the left.
  • Step 5(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