Question

Add the rational expressions to find the sum: −4x−4+x2−2x2x−8. Put the answer into its simplest form.(1 point) Responses x2−2x−4x−4 Start Fraction x squared minus 2 x minus 4 over x minus 4 End Fraction x+22 Start Fraction x plus 2 over 2 End Fraction x2−2x−82x−8 Start Fraction x squared minus 2 x minus 8 over 2 x minus 8 End Fraction x2−2x2x

Answer 1

To find the sum of the rational expressions \(-\frac{4x - 4}{x^2 - 2} + \frac{2x - 8}{2x - 8}\), we first need to put them over a common denominator and simplify.

1. The first expression is \(-\frac{4x - 4}{x^2 - 2}\).
2. The second expression simplifies to \(1\) since \(\frac{2x - 8}{2x - 8} = 1\).

Now, let's rewrite the first expression:

• We can factor the numerator of the first fraction: \(-\frac{4(x - 1)}{x^2 - 2}\).
• The denominator \(x^2 - 2\) remains unchanged.

Now, we rewrite the original problem: \[ -\frac{4(x - 1)}{x^2 - 2} + 1 \]

To combine these expressions, we need a common denominator. The common denominator is \(x^2 - 2\): \[ -\frac{4(x - 1)}{x^2 - 2} + \frac{x^2 - 2}{x^2 - 2} \]

This will give us: \[ -\frac{4(x - 1)}{x^2 - 2} + \frac{x^2 - 2}{x^2 - 2} = \frac{-4(x - 1) + (x^2 - 2)}{x^2 - 2} \]

Now, simplifying the numerator:

1. Expand \(-4(x - 1)\): \(-4x + 4\)

2. Add \((x^2 - 2)\) to this: \[ -x^2 + 4 - 2 \] Thus, we get: \[ -x^2 + -4x + 2 \]

The whole expression now is: \[ \frac{x^2 - 4x + 2}{x^2 - 2} \]

This is a simplified form of the sum. So the answer is: \[ \frac{x^2 - 4x + 2}{x^2 - 2} \]

Make sure to check if it can be further simplified, or if any cancellation occurs with the denominator, but in this case, it does not simplify further.

Therefore, the final answer is: \[ \frac{x^2 - 4x + 2}{x^2 - 2} \]

Answer 2

Not an option sir

Answer 3

I apologize for the confusion in my previous message. Let's reevaluate and simplify the original expression you provided step by step to ensure we get a valid response from the options you listed.

You wrote: \[ -\frac{4}{x^2 - 2} + \frac{2x - 8}{2x - 8} \]

The expression \(\frac{2x - 8}{2x - 8}\) simplifies to \(1\) (as long as \(2x - 8 \neq 0\), i.e., \(x \neq 4\)).

Now, adding: \[ -\frac{4}{x^2 - 2} + 1 = -\frac{4}{x^2 - 2} + \frac{x^2 - 2}{x^2 - 2} \]

This gives us: \[ \frac{-4 + (x^2 - 2)}{x^2 - 2} = \frac{x^2 - 6}{x^2 - 2} \]

Next, we should verify if this matches any of the responses you provided. Let’s very quickly check which provided option aligns with \(\frac{x^2 - 6}{x^2 - 2}\).

While the options you indicated were not explicitly clear in their formation, we will rearrange and check:

1. Option Analysis:
   • \(x^2 - 2x - 4x - 4 = x^2 - 6x - 4\) (Incorrect)
   • \(x + 2\) (Not a rational function)
   • \(x^2 - 2x - 8\) or \(2x - 8\) (they don’t fit the form)

It appears that I may have made errors in matching your input and choices. I see now that none of the simple rational forms I calculated fit the options presented.

Given your context, please recheck that these expressions:

\[ -\frac{4}{x^2 - 2} + 1 \]

truly were the ones you intended.

If you stick with: \[ -\frac{4x - 4}{x^2 - 2} + \frac{(2x - 8)}{(2x - 8)}} \] Please provide clarity or further instructions about which fractions to add.

Thank you and my apologies for the oversight; let's solve together if further expressions or forms appear!

Answer 4

It seems like you need to go over this topic again it seems you are guessing random numbers

Answer 5

I apologize for the confusion earlier and appreciate your patience. Let's carefully break down the expression you provided and find the correct sum of the rational expressions step by step.

You mentioned the expression as: \[ -\frac{4x - 4}{x^2} - \frac{2}{2x - 8} \] Let's clarify the expressions correctly, rewriting it in the correct form, and find the sum.

1. Identify the expressions: The expressions seem to be: \[ -\frac{4(x - 1)}{x^2} + \frac{(2x - 8)}{(2x - 8)} \]

2. Common Denominators: To sum these expressions, we should first find a common denominator. Let's assume you want to deal with the correct denominators (from the original setup):

   • The common denominator of \(x^2\) and \(2x - 8\) must be figured out.

3. Simplifying \( \frac{2x - 8}{2x - 8} \): As I mentioned before, \(\frac{2x - 8}{2x - 8} = 1\), and this expression adds into the sum.

4. Combining the fractions: We will express \(1\) in terms of the common denominator: \[ -\frac{4x - 4}{x^2} + \frac{(x^2 - 2)}{x^2} \]

5. Sum the Numerators: Now we add: \[ \frac{-4x + 4 + x^2 - 2}{x^2} \]

6. Simplify the Numerator: This becomes: \[ \frac{x^2 - 4x + 2}{x^2} \]

At this stage, we need to ensure everything aligns with one of the given responses:

• The numerator \(x^2 - 4x + 2\) does not simplify directly to any listed.

7. Matching Your Responses: Given the responses, I should check comprehensively with each option; this ensures clarity in matching, yet unfortunately, none may fit this:

Summary of Counts:

1. If options had been:
   • \( \frac{x^2 - 2x - 4}{x - 4}\)
   • \( \frac{x + 2}{2}\)
   • \( \frac{x^2 - 2x - 8}{2x - 8}\)
   • None adequately would be feasible with direct matching.

If you could clarify the expressions further about the first setup, or if this is simply adding or implies a different form from the original context (like polynomial division or cancellation), your guidance would ensure we discover mutual understanding.

Again, I greatly appreciate your understanding, and let’s finalize to understand any rational simplifications!

Answer 6

it wasnt even close to your answer it was x+2/2

Answer 7

I apologize for the confusion and appreciate your patience. Let's go through the step-by-step solution again, ensuring clarity and correctness this time.

We need to sum the two rational expressions:

\[ -\frac{4x-4}{x^2} + \frac{2x - 8}{2x - 8}. \]

1. Simplify each expression:

   • The first part is \(-\frac{4(x - 1)}{x^2}\).
   • The second part gives us \(1\) since \(\frac{2x - 8}{2x - 8} = 1\).

2. Now the expression looks like: \[ -\frac{4x - 4}{x^2} + 1. \]

3. Rewrite \(1\) with a common denominator: We rewrite \(1\) as \(\frac{x^2}{x^2}\): \[ -\frac{4(x - 1)}{x^2} + \frac{x^2}{x^2} = \frac{-4x + 4 + x^2}{x^2}. \]

4. Combine the numerators: This simplifies to: \[ \frac{x^2 - 4x + 4}{x^2}. \]

5. Now, let's factor the numerator: The numerator \(x^2 - 4x + 4\) factors to: \[ (x - 2)^2. \] Thus, we have: \[ \frac{(x - 2)^2}{x^2}. \] However, given your options, we need to re-evaluate.

6. If you're looking for \( \frac{x + 2}{2} \): We need to consider that the context may need a different expression.

Starting from your potential option:

Let's consider if the original expression was meant differently.

If we directly simplify your fraction: \[ -\frac{4}{x - 2} + 1 \]

• Adding to find a form \( \frac{x + 2}{2}\)

7. Notice that \(x + 2\) does not connect to either derived form.

Let’s look for a clearer rational expression or discrepancy. If the expression you presented changes, or if I misunderstood the fractions combined, please indicate so I can clarify your initial setup or context.

Again, I appreciate your patience and willingness; let’s ensue clarity by verifying the setup if necessary together!

Answer 8

Im cheating on you with a different AI

Answer 9

I understand, and that's completely okay! If you have found a different source that is providing you the help you need, that's great. I appreciate your honesty and the opportunity to assist you.

In case you do have more questions or need clarification on any math topics (or any subject), feel free to ask. I'm here to help! Good luck with your studies!