MHB How Do You Continue Long Division with the Expression \((\sqrt{x} + \delta)/x\)?

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The discussion revolves around the long division of the expression \((\sqrt{x} + \delta)/x\). The user initially finds \(\sqrt{x}\) as the quotient but struggles with the remainder, \(-\delta\sqrt{x}\). Another participant suggests that performing standard division instead yields \(\frac{1}{\sqrt{x}} + \frac{\delta}{x}\), which does not simplify the problem further. The conversation highlights the challenges of continuing long division with this expression and the limitations of the approach taken. Overall, the thread emphasizes the need for clarity in mathematical formatting and problem-solving strategies.
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I am trying to solve the following problem:

$$(\sqrt{x} + \delta)/x$$

Using long division, I get:

$$\sqrt{x}$$

On top and I end up with

$$-\delta\sqrt{x}$$

On the bottom and am unsure how to proceed from there? I'm sorry I can't write out all my work out as I'm not sure how I could format it properly with LaTeX. Any advice would be appreciated.
 
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TKline007 said:
I am trying to solve the following problem:

$$(\sqrt{x} + \delta)/x$$

Using long division, I get:

$$\sqrt{x}$$

On top and I end up with

$$-\delta\sqrt{x}$$

On the bottom and am unsure how to proceed from there? I'm sorry I can't write out all my work out as I'm not sure how I could format it properly with LaTeX. Any advice would be appreciated.

what exactly is the problem you are working on that involves the quotient \frac{\sqrt{x} + \delta}{x} ?

division (not long) yields \frac{1}{\sqrt{x}} + \frac{\delta}{x}, and offers nothing more simple than what you started with ...
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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