Certainly! Let's solve the equation [tex]\( Z = \frac{m}{\sqrt{x^2 - N^2}} \)[/tex] for [tex]\( x \)[/tex]. We'll go through this step-by-step.
1. Write down the original equation:
[tex]\[
Z = \frac{m}{\sqrt{x^2 - N^2}}
\][/tex]
2. Isolate the square root term:
Multiply both sides by [tex]\(\sqrt{x^2 - N^2}\)[/tex]:
[tex]\[
Z \sqrt{x^2 - N^2} = m
\][/tex]
3. Solve for the square root term:
Divide both sides by [tex]\( Z \)[/tex]:
[tex]\[
\sqrt{x^2 - N^2} = \frac{m}{Z}
\][/tex]
4. Square both sides to eliminate the square root:
[tex]\[
x^2 - N^2 = \left(\frac{m}{Z}\right)^2
\][/tex]
5. Simplify the right-hand side:
[tex]\[
x^2 - N^2 = \frac{m^2}{Z^2}
\][/tex]
6. Isolate [tex]\( x^2 \)[/tex]:
Add [tex]\( N^2 \)[/tex] to both sides:
[tex]\[
x^2 = \frac{m^2}{Z^2} + N^2
\][/tex]
7. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \pm \sqrt{\frac{m^2}{Z^2} + N^2}
\][/tex]
8. Simplify the expression under the square root:
[tex]\[
x = \pm \sqrt{N^2 + \frac{m^2}{Z^2}}
\][/tex]
Thus, the final expressions for [tex]\( x \)[/tex] are:
[tex]\[
x = \sqrt{N^2 + \frac{m^2}{Z^2}} \quad \text{and} \quad x = -\sqrt{N^2 + \frac{m^2}{Z^2}}
\][/tex]
