How to Isolate x in a Complex RLC Circuit Equation?

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Discussion Overview

The discussion focuses on isolating the variable x in a complex equation related to an RLC circuit. Participants explore the manipulation of the equation, including the application of the quadratic formula and considerations regarding potential extraneous solutions.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant presents an equation involving R, L, and C, suggesting a path to isolate x.
  • Another participant questions the presence of a square root of 2-1 but acknowledges the correctness of the approach.
  • A third participant introduces an alternative equation with a negative sign, indicating it may also yield a solution.
  • Participants emphasize the importance of checking all solutions against the original equation due to the squaring of both sides, which may introduce extraneous solutions.
  • Multiplying both sides by x is suggested as a method to derive a quadratic equation for x.

Areas of Agreement / Disagreement

Participants express different views on the manipulation of the equation and the implications of squaring both sides. No consensus is reached on the best approach to isolate x or the validity of all proposed solutions.

Contextual Notes

Participants note the potential for extraneous solutions arising from the squaring of the equation, which may affect the validity of the solutions derived.

p75213
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Can somebody point me in the right direction with this?

[tex]\begin{array}{l}<br /> \sqrt {{R^2} + {{\left( {xL - {\textstyle{1 \over {xC}}}} \right)}^2}} = \sqrt 2 R \\ <br /> {R^2} + {\left( {xL - {\textstyle{1 \over {xC}}}} \right)^2} = 2{R^2} \\ <br /> xL - {\textstyle{1 \over {xC}}} = R\sqrt {2 - 1} \\ <br /> \end{array}[/tex]
 
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I'm not sure why you have a square root of 2-1, but it is correct I guess.

From this point, multiply x on both sides and use the quadratic formula.
 
Don't forget that:

1. [itex]xL - {\textstyle{1 \over {xC}}} = -R\sqrt {2 - 1} \[/itex] (Note the minus sign for R) may yield a solution as well.

2. You should check all solutions you get in the original equation. Since your work involved squaring both sides of an equation, it is possible to have extraneous solutions.
 
Multiply both sides by x and you get a quadratic equation for x.
 

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