Completing the Square

Completing the square is a way to solve quadratic equations. Completing the square is generally not used much, thanks to the Quadratic Formula, which works better and saves you a lot of effort. However, it is still good to know this method, anyway. Besides, if you ask me, its fun! (I guess that's why I'm known as the math geek. :)

How to complete the square
Completing the square basically is creating a perfect binomial squared, and then square rooting everything else in the equation that doesn't make part of that binomial. The details are below.

1. Get the equation in standard form (ax^2+bx+c=0)

2. Divide every term in the equation by the "a" term (the coefficient of the x^2 term)

3. Subtract c (the constant) from both sides

4. Take "b" (the coefficient of the x term), divide it by 2, and square the result

5. Add the result found in step 4 to both sides of the equation.

6. Factor the perfect square trinomial you have just made

7. Square root both sides

8. Solve for x

That's it! Now do you see why everyone preferrs the quadratic formula? Alright, we'll take you through one example problem, and then we have some practice problems for you.

Example problem: Solve for x in 2x^2+4x-6=0
The equation is already in standard form, so we can move onto step #2. Dividing everything by 2, we get x^2+2x-3=0. Step #3 gives us the equation x^2+2x=3, after adding 3 to both sides.

Step #4 tells us to take the coefficient of the x term, divde it by 2, square it, and add the result to both sides. So, we take 2, divide it by 2 (1), square it (1), and add that result to both sides, getting x^2+2x+1=4. Now we can factor the left side into (x+1)^2=4. Perfect!

The rest is just inverse operations. Square root both sides (don't forget there are two solutions)and subtract 1, getting x=-1 +or- 2, which gets you the two solutions x=1 and x=-3.

Most answers do not come out as pretty as these. Generally, you will be dealing with square roots. Just don't be surprised when it happens.

Practice Problems
Boy! Aren't you excited! However, these are a good idea to do to make sure you understand everything - and this is a complicated, error-prone process. Try these out:

1. Solve x^2+6x-16 by completing the square.

2. Solve 3t^2-12t-15=0 by completing the square.

3. Solve -3t^2+12t-15=0 by completing the square.

4. Solve 5x^2-17x-13=0 by completing the square.

You can find the answers at the C page.