Completing the Square Lessons

Often, we encounter equations which cannot be easily solved by addition, subtraction, multiplication, division, and factoring. One such equation is

When the highest exponent of an equation is 2, the method of "Completing the Square" gives us an alternative. This method will help us turn this unfactorable equation into an equation that can be factored.

The Strategy

Consider the equation

We can solve this equation by simply taking the square root of each side.

This technique also works when we replace y with an expression like (p - 1):

Solving each of the resulting equations gives p = -2, 4.

The strategy used in completing the square is to get the square of a quantity equal to a number as in

Once this is done, create two subproblems as we did above.

The Process

The completing the square process has five major steps. The summary below assumes that the equation being solved is in the variable x.

1. Use addition and subtraction to move the constant term to the right and all other terms to the left.
2. Divide each term in the equation by the coefficient of the x² term, unless the coefficient is 1.
3. Determine the coefficient of the x term, divide it by two, square it, and add to both sides.
4. Factor the left side as a perfect square trinomial.
5. Take the square root of each side, and create two subproblems from the result.