With some quadratics, it's possible to look at them and -- at a glance, recognize that they are almost perfect squares. For example, consider the following:

x² + 6x + 8

If you've been working with quadratics for very long (or if you've done the previous reading in this unit), you know that x² + 6x + 9 is a perfect square - it's the square of (x + 3).

So what would we have to do to x² + 6x + 8 to turn it into a perfect square? That's easy! We'd just need to add 1 to it!

So let's see if we can formulate a method for completing squares. We know that (x + k)² = x² + 2kx + k², which should give us an idea...

If we take half of the coefficient of x and square it, we get the constant we would need in order to have a perfect square.

Example #1
What would we have to add to x² + 10x + 21 to make it a perfect square?

Solution #1
Take half of 10 (the coefficient of x), which is 5. Now square the 5 and you get 25. In order to be a perfect square, the constant term must be 25. Since it's actually 21, we need to Add 4 to complete the square.

Example #2
What would we have to add to x² - 4x + 5 to make it a perfect square?

Solution #2
Take half of -4, and you get -2. Square that and you get 4. Thus, the constant term must be 4. Since it's actually 5, we need to subtract 1 to complete the square.

But what if the coefficient of x² is something other than 1? Well, that changes it a bit. If the coefficient of x² is a perfect square, we can still complete the square easily. The rule is the same as before, except that now, after dividing the coefficient of x by 2, we also divide it by the square root of the coefficient of x². Then we square the result.

Example #3
What would we have to add to 9x² + 6x + 3 to make it a perfect square?

Solution #3
The coefficient of x² is 9, and its square root is 3. Divided the coefficient of x by 2 (that gives you 3), and then divide it by 3, which gives you 1. Square the 1, and you get 1. The constant term has to be 1, but it's actually 3, so we must subtract 2 to complete the square.

Example #4
What would we have to add to 4x² + 20x + 21 to make it a perfect square?

Solution #4
The square root of 4 is 2. Divide 20 by 2, and then divide that by 2, and you get 5. Square the 5, and we find that our constant term must be 25. Since it's actually 21, we need to add 4 to complete the square.

For each problem below, find what must be added or subtracted to complete the square: