In this blog post, I will be attempting to explain how to solve a quadratic equation using the method of completing the square. Out of the three methods, I personally find completing the square to be the most difficult, but everyone has their own personal preference.

We will be solving the equation from textbook page 209 as an example.

This equation is simpler than others as the numbers are factors of 3

Step #1 – To solve this, we must change the first term from to this is always the case when using this method. We will also divide by 3.

Step #2 – Write the equation in a factored form. To ensure it is correct you can re-distribute. If you receive the same equation as the original question you have done this correctly. In this case, it should be

Step #3 – When using the method of completing the square, you take the middle term, and use the formula seen in the photo. In this case, it will be   (The squaring applies to both the 6 and the two)

Step #4 – After step 3, the formula should have given you 9. This is not always the case, however, since sometimes your numbers will not divide evenly into each other. In that case, leave it in fraction form. Now it should look like

Step #5 – Remove the negative 9 from the brackets. To do this, multiply the 3 by the negative 9 as shown in the photo. Then once it is removed, simplify the right side and move the -29 past the equals sign, as shown in the photo. This will give you

Step #6 – To finally get rid of that 3 in front, divide by 3. This means the 3 disappears from the front of the equation, and 29 becomes

Step #7 – Almost there – To remove the square from the equation, you must square root it, and what you do to one side has to be done to the other.So:  

Step #8 – We’re almost done, but root three is irrational, so we have to fix that. That is done by multiplying both the numerator and denominator by the denominator. In this case, it will be root 3. It should become   (the root does not apply to the three, since we’ve rationalized it.

Step #9 – Now, all you’ve gotta do is move the 3 over to the right side, giving you the answer! yay!

If you’ve read this far I hope I’ve helped you understand this concept, remember that you always must include a “plus-minus” sign before a root, since the answer could be both positive or negative.