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Solving Quadratic Equations:
     Solving by Completing the Square (page 3 of 6)

Sections: Solving by: factoring, taking roots, completing the square, using the formula, graphing

The quadratic in the previous section's last example, "(x – 2)² – 12", can be multiplied out and simplified to be "x² – 4x – 8". We would not have been able to solve the equation with the quadratic formatted this way because it doesn't factor and it isn't ready for square-rooting. The only reason we could solve it before was because they'd put all the x stuff inside a square, so we could square-root both sides. So how do you go from a regular quadratic like "x² – 4x – 8" to one that is ready to be square-rooted? We would have to "complete the square".

{I have a lesson on solving quadratics by completing the square, which explains the steps and gives examples of this process. It also shows how the Quadratic Formula is generated by the completing-the-square process. So I'll just do just this one example here. If you need further instruction, read the lesson at the above hyperlink.)

• Use completing the square to solve x² – 4x – 8 = 0.

As we noted before, this does not factor, so we can't solve the equation that way. And they haven't given us the quadratic in a form that is ready to square-root. But there is a way to manipulate the quadratic to put it in a form that we can square-root. It works like this:

First, you put the loose number on the other side:

x² – 4x – 8 = 0
x² – 4x = 8 Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Then you look at the coefficient of the x-term; this is –4, in this case. Take half of this (including the sign); this is –2. Then square this value to get +4, and add this to both sides:

x² – 4x + 4 = 8 + 4
x² – 4x + 4 = 12

This process created a perfect-square quadratic on the left-hand side. That is, if I factor the left-hand side, I get a square:

(x – 2)² = 12

Now I can square-root both sides and solve:

(x – 2)² = 12



Then the solution is   .

Note: In general, unless you're told that you have to use completing the square, you will not use this method. Either some other method (like factoring) will be quicker, or else the Quadratic Formula (coming up next) will be easier. However, if you have covered completing the square in class, you should expect to see a problem or two of this type on the next test, and maybe one on the Final. But you can probably safely forget this process after that.

However, not needing to remember completing-the-square is not the same as not having problems with messy answers like the above. But instead of completing the square to find the messy solutions, you'll use a formula.

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