How to Isolate x in a Complex RLC Circuit Equation?

[p75213]

Can somebody point me in the right direction with this?

$$\begin{array}{l} \sqrt {{R^2} + {{\left( {xL - {\textstyle{1 \over {xC}}}} \right)}^2}} = \sqrt 2 R \\ {R^2} + {\left( {xL - {\textstyle{1 \over {xC}}}} \right)^2} = 2{R^2} \\ xL - {\textstyle{1 \over {xC}}} = R\sqrt {2 - 1} \\ \end{array}$$

 

[scurty]

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.

 

[Redbelly98]

Don't forget that:

1. $xL - {\textstyle{1 \over {xC}}} = -R\sqrt {2 - 1} \$ (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.

 

[HallsofIvy]

Multiply both sides by x and you get a quadratic equation for x.

 

[CellCoree]

To solve this equation for x, we need to isolate the variable x on one side of the equation. We can do this by first simplifying the equation and then using algebraic operations to manipulate it.

1. Simplify the equation by expanding the squared terms on both sides:
R^2 + x^2L^2 - 2xL/C + 1/x^2C^2 = 2R^2

2. Combine like terms on the left side of the equation:
x^2L^2 - 2xL/C + 1/x^2C^2 = R^2

3. Move all terms with x to one side of the equation by subtracting R^2 from both sides:
x^2L^2 - 2xL/C + 1/x^2C^2 - R^2 = 0

4. Factor out x from the first two terms on the left side:
x(xL^2 - 2L/C) + 1/x^2C^2 - R^2 = 0

5. To simplify the equation further, we can multiply both sides by x^2C^2:
x(xL^2 - 2L/C) + 1 - R^2x^2C^2 = 0

6. Use the quadratic formula to solve for x:
x = (2L/C ± √(4L^2/C^2 - 4(1-R^2C^2)))/2

7. Simplify the expression inside the square root:
x = (2L/C ± √(4L^2/C^2 - 4 + 4R^2C^2))/2

8. Simplify further and solve for x:
x = (2L/C ± √(4L^2/C^2 + 4R^2C^2 - 4))/2
x = (2L/C ± 2√(L^2/C^2 + R^2C^2 - 1))/2
x = L/C ± √(L^2/C^2 + R^2C^2 - 1)

Therefore, the solutions for x are:
x = L/C + √(L^2/C^2 + R^2C^2 - 1)
x = L/C - √(L^2