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Completing the square usually starts with gathering the variable terms on one side of the equation and the constant terms on the other side. Your equation is already in this form.
The -4 in front of the squared term makes the problem a little more difficult. We can proceed by factoring out the -4 or by dividing both sides by the -4. Either way we will have some difficulties. I'll do it both ways so you can see this. I'll start by factoring out the -4:
We want a perfect square trinomial in the parentheses. Since the coefficient of the squared term is a 1 (since the -4 is now outside the parentheses) all we have to do is:
Find 1/2 of the coefficient of the x term. (Remember this number because we will use it again later.)
Find the square of the number found in the previous step.
Add the number found in step 2 to each side. (Note: this can be tricky to get right if the perfect square trinomial has a factor in front of it.)
Rewrite the perfect square trinomial in the form: where "n" is the number you found in step 1.
Let's see this in action:
1. Find 1/2 of the x coefficient.
The coefficient of x is -4. Half of this is -2.
2. Square the number from step 1.
The square of -2 is 4
3. Add the number from step 2 to both sides.
Because of the -4 in front this will be tricky. We want the left side to be:
where the expression in the parentheses is the perfect square trinomial we've been building. When you add something to one side of an equation you must balance it by adding the same thing to the other side. So, looking at our left side, what did we add to it when going from
to
The "obvious" but wrong answer is 4!? The +4 was added inside the parentheses. Because of the -4 in front, what we really added was -4*4 or -16! So it is -16 that we must add to the right side! Adding the "+4" properly gives us:
4. Rewrite the trinomial as a perfect square.
Using the number from step 1 we get:
or simply
The square has been completed. We can now solve. First we'll divide by -4:
At this point we should recognize a problem. We have a perfect square on the left. And the equation says this perfect square is equal to a negative number. If you are only interested in real number solutions, then there are none for this equation and we can stop here.
There are only complex solutions. If you want the complex solutions then our next step is to find the square root of each side:
Because of the fact of positive and negative square roots we end up with two equations: and
Simplifying the right sides we get: and and and
Last we add 2 to each side: and
These are the complex solutions to your equation. If you want them in standard a + bi form they would be: and
P.S. Now I will show you how this is done when you divide by -4 at the beginning.
Dividing by -4:
As you can see, the difficulty with this method is that we get to work with fractions for most of the problem. Next we complete the square. The steps (and numbers are the same as before except when we add the "+4" we are actually adding a 4 (since there is no factor in front)!
On the left we get our perfect square. On the right we have to get common denominators to add:
which simplifies to:
And the rest is the same as the earlier solution.
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