How to Solve a Quadratic Equation by Completing the Square

Completing the square is one of the most frustrating and easily forgettable algebraic processes. The quadratic formula is complex but straightforward and easy to use once memorized, so why even bother using "completing the square" to find the solutions to a quadratic equation? Besides the obvious answer that the teacher is probably requiring it, completing the square is useful for other things in algebra such as putting a parabola in vertex form or writing the equation of an eclipse or hyperbola. Therefore, it is good to practice these steps.

Difficulty: Moderate

Instructions

1. 

   Make sure the leading coefficient is 1. This is the number in front of the X squared. If it isn't 1, divide everything in the problem by the leading coefficient. For instance, if you have 3 X squared, divide everything by 3. If you get fractions, don't panic! You will just have to take your time to work with fractions through the problem.

2. 

   Move the constant (the plain number with no X in it) over to the right side by adding or subtracting it from both sides. Do the opposite of whatever is in the original equation.

3. 

   Find "b." This is the number in front of the plain X, not the X squared. Write "b" down off to the side. Make sure you include a negative sign if there is one in front of the "b". Then divide it by two and circle. Than square that and put a box around your answer.

4. 

   Add the number in the box to both sides of the equation.

5. 

   Now you are going to completely rewrite the left side of the equation. You are basically saying what could I foil (multiply out) that would make what I have now? Luckily, completing the square tells you how to do this so you don't have to figure it out. Draw parenthesis. Inside the parenthesis, write "X + your circled number" if the number is positive or "X - your circled number" if it is negative. Then put a "squared" on the outside of the parenthesis.

6. 

   Now you can take the square root of both sides to solve for X. Because you are putting a square root sign in, you need to have two equations. One with a positive answer and one with a negative.

7. 

   If you are able to take a square root of the right side, you will end up with two integer answers as in this problem. If you cannot take the square root you need to simplify the radical in both equations and finish solving for X by adding/subtracting the remaining number on the left to both sides.

Tips & Warnings

• Make sure you write down all three: b, b/2, and b squared and keep track of them.
• If you are still having trouble with completing the square in Algebra, try finding a math tutor to explain it to you. Sometimes hearing the steps is easier than seeing them. See the resources below for a guide to picking the best math tutor.

Resources

• How to Get Math Homework Help
• How to Select a Good Math Tutor

[Gottaloveit]

on 6/11/2009 Heck, I graduated college and have NO idea what a quadratic equation is (biology degree). But, an interesting read anyway!

[gahazeleyes]

on 6/1/2009 Wow, now you took me back. I remember this in college. Thanks for a different well needed article.