Completing the Square for Conic Sections

By Yang Kuang, Michelle Rose Gilman

When the equation of a conic section isn’t written in its standard form, completing the square is the only way to convert the equation to its standard form. The steps of the process are as follows:

  1. Add/subtract any constant to the opposite side of the given equation, away from all the variables.

  2. Factor the leading coefficient out of all terms in front of the set of parentheses.

  3. Divide the remaining linear coefficient by two, but only in your head.

  4. Square the answer from Step 3 and add that inside the parentheses.

    Don’t forget that if you have a coefficient from Step 2, you must multiply the coefficient by the number you get in this step and add that to both sides.

  5. Factor the quadratic polynomial as a perfect square trinomial.