Question

Solve the equation by completing the square: x^2 = 24 - 4x

Answer 1

Let's rearrange this a bit:
x^2 + 4x + 4 = 28. Do you know what to do next?

Answer 2

do i distribute the 4 to the other side to get x^2+4x=24

Answer 3

Well, no, that would almost bring you back to the starting point. Do you know what does "completing the square" mean?

Answer 4

sorta, not really

Answer 5

Ok, so from the beginning. From an equation, you want to get x. But when it's a quadartic equation, you have not only x, but also x^2. When you have something squared, it would be natural to take a square root of it. When you have for example x^2 = 4, it works. But in this case, it won't, because the value sqrt(x^2+4x) (that you would get if you squared both sides) doesn't tell you much about the value of x itself. Luckily, you have this helpful equation: (a+b)^2 = a^2 + 2ab + b^2.

Answer 6

So when you have a more complicated quadratic equation, like yours, you basically want to transform x^2+4x into a square of something, so that you can take a square root of it. It means you go from the right-hand side of this equation, to left-hand side of it. Ok, let's take care of your example:
x^2 + 4x - well, we have our a^2 here and 2ab. It suggests that our square should be (x+2)^2. It is equal to x^2 + 4x + 4 => you need to get this expression into the left side of the equation, and everything else into the right side.
So you get this:
x^2 + 4x + 4 = 28. (Do you see how this is equivalent to your original equation?)

Answer 7

And further: so you can transform it, as I've said previously.
(x+4)^2 = 28
Can you solve it now? That method is called "completing the square".

Answer 8

yes, i understand up to that point, but the rest i dont.. I have learned this like last year but i dont remeber it anymore so im like lost. and my textbook online dosent really show me anything.

Answer 9

Sorry, where did you get lost?

Answer 10

before your last post

Answer 11

like i understand up until (x+4)^2 = 28

Answer 12

Yeah, right, and this is basically the end. Now it's like you had x^2=4 - you can take a square root of both sides.
\[\sqrt{(x - 4)^2} = \sqrt{28}\]
\[x - 4 = \pm \sqrt{28}\] (remember that there are two solutions!)
\[x = \sqrt{28} + 4\] or \[x = -\sqrt{28} + 4\]. And that's the answer.