Question

What is the Least Common Denominator? Screenshot attached!

Answer 1

Its 2

Answer 2

Oh sry dont have time...

Answer 3

Haha sorry

Answer 4

In the second term on the left, x^2 - 4 can be factored into (x + 2) and (x - 2)

Answer 5

(x-2)*(x+2)

Answer 6

:D

Answer 7

yes

Answer 8

x^2-4=(x+2)(x-2)

on the other hand you have (x+2) so you need to multiply den and num by (x-2)

Answer 9

so you get it

Answer 10

\[x^2-4\] is the least common multiple of your denominators

Answer 11

I'm here and can help.

Answer 12

Just multiply the first term on the left by (x-2) / (x-2). Leave that factor in both the numerator and the denominator for now and then put the "+" and the second term on the left. For that second term on the left, factor out the denominator as in my previous post.

Answer 13

explanation is the that
\[x^2-4=(x^2-4)\times 1\]
\[x^2-4=(x+2)(x-2)\]
\[x^2-4=(x-2)(x+2)\] so it is a multiple of each denominator

therefore the easy way to solve this equation is to multiply each side by \(x^2-4\) so clear the fractions

Answer 14

Your new numerator on the left will be a COMBINED numerator of 3(x-2) + 12.

Answer 15

x²-4 = (x+2)(x-2)

so the least common denominator

Answer 16

is x²-4

Answer 17

Once you simplify the new COMBINED numerator, gotten by using the least common denominator, you can factor the new numerator and one of the factors in the denominator will cancel.

Answer 18

\[(x^2-4)\times \left(\frac{3}{x+2}+\frac{12}{x^2-4}\right)=\frac{-1}{x-2}\times (x^2-4)\]

Answer 19

3(x-2) + 12 will become 3x + 6 which will become 3(x + 2) and that second factor will cancel.

Answer 20

\[3(x-2)+12=-1(x+2)\] after cancelling
solve for \(x\) now is easier

Answer 21

I jumped in because others left for a while. If you want to use LCM, better to use my method as it illustrates LCM more straightforward. Other helper's methods here are a little different from mine, so you pick who you want to listen to at this point. The important thing is that you get the concept, so you are free to choose.

Answer 22

@tcarroll010 ?

Answer 23

You're sort of on the right track, more on than off. I think it is best that I start you over a little bit. BTW, loved your "last man standing", though not exactly a confidence-booster! The problem is really two-fold. You are asked to get LCD, but you have an equation, which is actually solved for you already. The underlying purpose for this is to see how they got the answer (the right-hand side). So, I'll post some stuff for you to consider.

Answer 24

[3(x-2)] / [(x+2)(x-2)] + 12 / [(x+2)(x-2)] is the left side. x^2 - 4 is the LCD and I also factored that out in the second term at the same time.

Answer 25

I'm floating now. All is peachy.

Answer 26

This is fun because you have personality. Yes, I would suggest this as a "starting over" so you don't have to read and re-read. Simpler is better.

Answer 27

For the COMBINED numerator, 3(x - 2) + 12 = 3x - 6 +12 = 3x + 6 = 3(x + 2) and THAT is the second trick to this problem. The first was getting the LCD. From that point, you can cancel out the (x+2) from the numerator and the denominator.

Answer 28

Getting the LCD and simplifying the left side is one thing. Once you have that, then we need to talk about the equality and the right side.

Answer 29

x first must be different of 2 and -2
3/x+2 + 12/x²-4 = -1/x-2
=>[ 3(x-2)+12]/x²-4 = -1(x+2)/x²-4
we eliminate x²-4 on each side and we got 3(x-2)+12 = -x-2
so 3x-6+12=-x-2
then 4x=-8
so x=-2
IMPOSSIBLE this equation has no solution ; cause x must not be 2 or -2

Answer 30

we can never devide by 0 in maths

Answer 31

Aylen33, I was getting to that, but step-by-step so as not to just give the answer to the student. The students have to show some work here.

Answer 32

oh yah sorry for that may be she will just get the method first ;)

Answer 33

Once you simplify the left-hand side, you will end up with the strange-looking 3 / (x - 2) = (-1) / (x-2). These two expressions on either side of the "=" sign have the same denominator, so we need only look to the numerator to test the equality. Notice anything strange?

Answer 34

wow this is a lot of work

Answer 35

Aylen333 brought up a good point about what are acceptable values for x when looking at the denominator of the LCD, but that becomes secondary as the "equality" is just a nonsense statement. However, the question just asked for finding the LCD. Discovering that the "=" doesn't hold is just a result of further investigation.

Answer 36

So, even though the equality doesn't hold, we just have to settle for having found the LCD, which is what the denominator of the second term on the left is. And then we're done, and don't really have to try to "solve".

Answer 37

So, in a nutshell, x^2 - 4 is LCD and that's it.

Answer 38

greeting from north africa

Answer 39

Sometimes we make things more complicated when we don't have to. But sometimes it is good to take things "a step further", like here to see if there is a solution.

Answer 40

So, all good now?

Answer 41

For the other helpers, thanks for pitching in. @satellite73 , your method was good but needed to mention that x = -2 won't work, because you multiplied that out of the denominator when in reality it has to be "checked" for allowable values.

Answer 42

Same with x = 2.