Simplify each complex fraction. [ 1/[(x+h)^2 + 9] - 1/(x^2+9)] / ...

Question

Simplify each complex fraction. [ 1/[(x+h)^2 + 9] - 1/(x^2+9)] / h

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we want to simplify the expression three divided by X plus a squared plus three minus three divided by X squared plus three. And all of that is being divided by a. So in this case you can see that we're simplifying a expression with a lot of components. So in order to make it easier to simplify, I'm only going to look at the numerator to start with. So if I look at the numerator I have three divided by X plus a squared plus three minus three divided by X squared plus three. So in this case we're working with two fractions but they have different denominators so we can't combine them but we can make them have the same denominator in order to combine them. We just need to multiply by the other fractions denominator. So this first fraction I need to multiply with X squared plus three and recall that I need to multiply by X squared plus three over X squared plus three. Because this is one. If I simplify it. So I'm not introducing any new numbers into my expression. And for my second fraction I need to multiply by X plus a squared plus three. Again needs to be equivalent to one. So we need to put it over itself. And now if I do this multiplication I have three times x squared plus three divided by X squared plus three times X plus a squared plus three minus three times X squared X plus a squared plus three divided by X squared plus three times X plus a squared plus three. And from here you can see that we have the same denominator for our fractions So we can combine them and once I combine them it becomes three times X squared plus three minus three times X plus a squared plus three divided by X squared plus three times X plus a squared plus three. And sorry, you want to make sure to have the correct fractions here or parentheses? Because this is the one of the denominators, there's a fraction around X plus a squared plus three. So you want to make sure to include that so that the X squared plus three gets distributed to each of those components. But now that we have combined our fractions and we know the common denominator, I'm going to focus on simplifying this numerator here. So if I focus on the numerator, I have three times X squared is three X squared and three times positive three is minus three times X plus a squared. So I'm gonna go ahead and expand that. So I have X plus a times X plus a plus three. And now I can do the foil method to expand these here. So I'll still have three X squared plus nine minus three and now I have X times X is X squared X times A is a X A times X is again a X and eight times A is a squared. We still have the plus three. And you can see I can combine these middle terms A. X plus A. X. Is to A X. So I can rewrite this as three X squared plus nine minus three times X squared plus two A X plus a squared plus three. And now everything is fully expanded. So I can go ahead and multiply this negative three. Three X squared plus nine negative three times X squared is negative three X squared negative three times to A. X. Is negative six. A. X negative three times A squared is negative three A squared and negative three Times positive three is -9. And from these you can see that three X squared cancels out with negative three X squirt And positive nine cancels out with -9. So we're left with just negative six A. X minus three A squared. And if we look at these two components, you can see that I can actually pull out a negative three A. So if I pull out negative three A. From this expression I'm left with two x plus a. But remember that this is the numerator in this expression. So if I rewrite this back together, I have negative three A times two X plus a over X squared plus three times X plus a squared plus three. And you might think that we're done. But if we look back, recall that all of that was over A. So all of this is divided by A. You can see so because I have a fraction on top of another fraction, I can write this in the more linear format which would be negative 38 times two X plus A divided by X squared plus three times X plus a squared plus three. And that's being divided by a over one. Well recall that I can change this into a multiplication. If I switch the numerator and denominator of the second fraction. So instead of being a over one I now have one over A. And I can now cancel this A. With this A. In the numerator. So that leaves me with negative three times two. X plus A divided by X squared plus three times X plus a squared plus three. And I can't simplify this any further. So this becomes my final simplification for the expression. So if we look at the answer choices, you can see that answer choice B also has negative three times two X plus A. Divided by X squared plus three times X plus a squared plus three. So B is our final answer. So hopefully this video helped and see you next time

Simplify each complex fraction. [ 1/[(x+h)^2 + 9] - 1/(x^2+9)] / h

Key Concepts

Complex Fractions

Limit Definition

Algebraic Manipulation

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