\( \frac { \sqrt { a x - p } } { a + b x } \)

Here is the expression you've provided: \[ \frac{\, \sqrt{a x - p} \,}{\, a + b x \,} \] ### Breakdown of the Expression: - **Numerator:** \(\sqrt{a x - p}\) - This represents the square root of the linear expression \(a x - p\). - **Denominator:** \(a + b x\) - This is a linear expression in \(x\). ### Possible Operations You Might Consider: 1. **Simplification:** - The expression is already in its simplest form unless there are specific values or relationships between the constants \(a\), \(b\), and \(p\). 2. **Differentiation:** - If you need to find the derivative with respect to \(x\), you can apply the quotient rule combined with the chain rule. \[ \frac{d}{dx} \left( \frac{\sqrt{a x - p}}{a + b x} \right) = \frac{ \frac{a}{2\sqrt{a x - p}} \cdot (a + b x) - \sqrt{a x - p} \cdot b }{(a + b x)^2} \] 3. **Integration:** - Integrating this expression with respect to \(x\) might require substitution or other integration techniques. For example, let \(u = a x - p\), then \(du = a\,dx\), and you can rewrite the integral in terms of \(u\). 4. **Solving for \(x\):** - If you need to solve the equation \(\frac{\sqrt{a x - p}}{a + b x} = 0\), you would set the numerator equal to zero: \[ \sqrt{a x - p} = 0 \implies a x - p = 0 \implies x = \frac{p}{a} \] This is the only solution, provided that \(a x - p \geq 0\). 5. **Graphing:** - Plotting the function can help visualize its behavior based on different values of \(a\), \(b\), and \(p\). ### Let Me Know If You Need Further Assistance If you have a specific operation or analysis in mind for this expression—such as simplifying, differentiating, integrating, or solving for certain variables—please let me know, and I'd be happy to help further!