Simple square root addition question

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Discussion Overview

The discussion revolves around simplifying expressions involving square roots and solving equations that include them. Participants explore methods for manipulating square root terms, particularly in the context of physics problems related to momentum and subatomic particles.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant presents a square root expression and requests step-by-step assistance in simplifying it.
  • Another participant suggests expanding the squares of sums and differences as a potential method for simplification.
  • A different participant introduces a technique for simplifying the difference of two square roots, emphasizing its importance in avoiding significant digit loss during calculations.
  • One participant shares a related problem involving momentum and square roots, asking how to solve for a variable.
  • Several participants discuss the process of squaring both sides of an equation to eliminate square roots, leading to a quadratic equation.
  • Another participant expresses uncertainty about the simplification process after squaring an equation.
  • One participant claims to have found a solution for a variable but later realizes that the example was made up and has no real solution.
  • Another participant suggests testing a different example to further explore the concepts discussed.
  • Participants share approximate values for the variable in question, with some expressing doubts about the correctness of their solutions.
  • There is a suggestion to substitute an imaginary number into the equation, indicating a potential misunderstanding of the problem's nature.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct solutions for the variable in the equations discussed. There are multiple competing views regarding the simplification methods and the nature of the solutions, particularly concerning real versus imaginary results.

Contextual Notes

Some participants express uncertainty about the simplification steps and the implications of their results, indicating that the discussion may depend on specific assumptions or definitions not fully explored.

peppies
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I know this must be an easy question, but I can't seem to remember how to do it:

\sqrt{z^2+R^2 - 2zR} - \sqrt{z^2+R^2 + 2zR}

Can someone go through step by step how to solve this? This isn't a homework question but I've been running into this problem more often in multiple courses.

So far, I've gotten down to something like this but I'm not sure if this is the right track. I replaced parts of the above equation to make it simpler but I STILL can't remember how to do it:

\sqrt{C^2 - x} - \sqrt{C^2 + x}

Thanks
 
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Expand these expressions:
(A + B)² = ?
(A - B)² = ?
 
Simplify even further, and utilize the trick:
a-b=(a-b)\frac{a+b}{a+b}=\frac{a^{2}-b^{2}}{a+b}
This type of simplification can be necessary prior to performing finite-digit arithmetic on the computation (i.e, by a computer), since otherwise, there might result a loss of significant digits when subtracting almost equally large numbers from each other.
 
Doc Al said:
Expand these expressions:
(A + B)² = ?
(A - B)² = ?
Oops, I forgot those two..:shy:
 
Doc Al, thanks for the hint! I was able to figure it out now.

I do have another similar problem in another course with unequal variables. For instance, I have to calculate the momentum of some subatomic particles and I get an equation like:

1000 =\sqrt{(pc)^2 + 875000} + \sqrt{(pc)^2 + 130000}

How would I solve for "pc" in this case?
 
No easy trick here, I'm afraid. Just get rid of those square roots. Call (pc)² = x (to keep it simple) and square both sides. Then rearrange and do it again. (You'll end up with a quadratic to solve.)
 
Doc Al said:
No easy trick here, I'm afraid. Just get rid of those square roots. Call (pc)² = x (to keep it simple) and square both sides. Then rearrange and do it again. (You'll end up with a quadratic to solve.)
What are you doing in Maths, Doc?
 
Hootenanny said:
What are you doing in Maths, Doc?
I took a wrong turn somewhere... :rolleyes:
 
Thanks Doc Al and arildno for the quick responses.

Do you know what the solution is for "pc" in that example above? I am still in the process of trying to solve that one but my skills are a little rusty.

So if you square both sides, do you get:

1000^2 = [\sqrt{(pc)^2 + 875000} + \sqrt{(pc)^2 + 130000} ]^2

I am having trouble in trying to simplify the right side.
 
  • #10
Use this "trick" again: (A + B)² = ?
 
  • #11
Super! It took awhile but at least I got this down pretty good again. I did some searching online to refresh my square root concepts and solve the rest. I solved a problem in the book that I knew the answer to, but the problem above is one that I just made up randomly so I didn't know the answer.

I got pc = 337 approximately. If you solved this, let me know if you got the same thing.

Thanks
Regards
 
  • #12
peppies said:
but the problem above is one that I just made up randomly so I didn't know the answer.
I'm glad that this is just a made up example, since it has no real solution (only an imaginary one)! (I didn't realize that until I tried to solve it myself.)

Pick a different example to test your skills:

1000 = √[(pc)^2 + 1000] + √[(pc)^2 + 2000]
 
  • #13
I was able to get about 498.5 roughly. Is this correct?
 
  • #14
Actually, all I would just have to do is plug that answer back into the equation and it looks like it is correct.

Thanks for the help Doc Al
 
  • #15
peppies said:
I was able to get about 498.5 roughly. Is this correct?
Looks good.
peppies said:
Actually, all I would just have to do is plug that answer back into the equation and it looks like it is correct.
Now you're thinking. :approve:
Thanks for the help Doc Al
You are welcome.
 
  • #16
peppies said:
I got pc = 337 approximately. If you solved this, let me know if you got the same thing.

If I substitute pc=337 into the equation of post #5, I don't get 1000.

Try this: what value of pc gives a minimum for the right-hand-side expression? How does that value compare with the 1000 on the left-hand-side?
 
  • #17
Redbelly98 said:
If I substitute pc=337 into the equation of post #5, I don't get 1000.
Try substituting 337i. :wink:
 
  • #18
Doc Al said:
Try substituting 337i. :wink:

Much better. I will sleep well tonight.
 

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