$\frac{x-2}{\sqrt{x^2-4}}$

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06-11-2026
Mathematics

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$\frac{x-2}{\sqrt{x^2-4}}$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T01:22:57-05:00

The domain of the expression $\frac{x-2}{\sqrt{x^2-4}}$ is $x \in (-\infty, -2) \cup (2, \infty)$.
For $x > 2$, the expression simplifies to $\sqrt{\frac{x-2}{x+2}}$.
For $x < -2$, the expression simplifies to $-\sqrt{\frac{x-2}{x+2}}$.
The simplified form is $\text{sgn}(x) \sqrt{\frac{x-2}{x+2}}$, where $\text{sgn}(x)$ is the sign of $x$.
The final answer is $\boxed{\frac{x}{|x|} \sqrt{\frac{x-2}{x+2}}}$.

Explanation

Domain and Analysis of the Expression

The expression is $f(x) = \frac{x-2}{\sqrt{x^2-4}}$. To determine the domain, the denominator must be non-zero and the radicand must be positive, so we require $x^2 - 4 > 0$. Solving this inequality, we find $x^2 > 4$, which implies $x \in (-\infty, -2) \cup (2, \infty)$. We analyze the expression in two intervals: $x > 2$ and $x < -2$.

Simplification for $x > 2$

For $x > 2$, the expression is positive. We can factor the denominator as $\sqrt{(x-2)(x+2)} = \sqrt{x-2}\sqrt{x+2}$. Since $x > 2$, $x-2 > 0$, so we can write $x-2 = (\sqrt{x-2})^2$. Thus, the expression becomes:
\[ \frac{x-2}{\sqrt{x^2-4}} = \frac{(\sqrt{x-2})^2}{\sqrt{x-2}\sqrt{x+2}} = \frac{\sqrt{x-2}}{\sqrt{x+2}} = \sqrt{\frac{x-2}{x+2}} \]
This holds for all $x > 2$.

Simplification for $x < -2$ and Final Summary

For $x < -2$, the expression is negative because the numerator $x-2$ is negative and the square root is positive. We factor the denominator as $\sqrt{x^2-4} = \sqrt{(x-2)(x+2)}$. Since $x < -2$, both $x-2 < 0$ and $x+2 < 0$, so their product is positive. We can write the numerator as $-(2-x)$. Note that for $x < -2$, $2-x > 0$. The expression becomes:
\[ \frac{x-2}{\sqrt{(x-2)(x+2)}} = \frac{-(2-x)}{\sqrt{(2-x)(-x-2)}} = -\frac{\sqrt{2-x}\sqrt{2-x}}{\sqrt{2-x}\sqrt{-(x+2)}} = -\frac{\sqrt{2-x}}{\sqrt{-x-2}} = -\sqrt{\frac{2-x}{-x-2}} \]
Distributing the negative sign in the fraction or simplifying, we observe this is equivalent to:
\[ -\sqrt{\frac{x-2}{x+2}} \]
Thus, the expression simplifies to $\sqrt{\frac{x-2}{x+2}}$ for $x > 2$ and $-\sqrt{\frac{x-2}{x+2}}$ for $x < -2$. These cases can be summarized using the signum function as $\text{sgn}(x) \sqrt{\frac{x-2}{x+2}}$ (specifically, for $x > 2$, $\text{sgn}(x) = 1$, and for $x < -2$, the sign is negative). A more standard way to write this is $\frac{x}{|x|} \sqrt{\frac{x-2}{x+2}}$.

Examples

This type of algebraic simplification is essential in calculus, especially when dealing with limits or derivatives involving square roots. For instance, when evaluating the limit of a function like $f(x) = \frac{x-2}{\sqrt{x^2-4}}$ as $x$ approaches 2 or infinity, simplifying the expression helps identify the behavior of the curve. Understanding how to handle domains and signs of square roots ensures accuracy when modeling real-world phenomena such as rates of change in physics or finance.