Let's start by carefully analyzing the equation you've provided and solving for $p$. The equation is:

$\frac{1}{x^2} - p + \frac{e^{\ln(1)}}{16} = \left(\frac{2}{x} - \frac{1}{4}\right)^2 + \sin(x)$

Step 1: Simplify the Exponential Term

First, let's simplify $\frac{e^{\ln(1)}}{16}$: $e^{\ln(1)} = 1 \quad \text{(since } \ln(1) = 0 \text{ and } e^0 = 1\text{)}$ Thus, $\frac{e^{\ln(1)}}{16} = \frac{1}{16}$

Step 2: Substitute Back into the Equation

Now, the equation becomes: $\frac{1}{x^2} - p + \frac{1}{16} = \left(\frac{2}{x} - \frac{1}{4}\right)^2 + \sin(x)$

Step 3: Expand the Squared Term

Expand $\left(\frac{2}{x} - \frac{1}{4}\right)^2$:

= \frac{4}{x^2} - \frac{1}{2x} + \frac{1}{16}$$ ### Step 4: Substitute Back into the Equation The equation now is: $$\frac{1}{x^2} - p + \frac{1}{16} = \frac{4}{x^2} - \frac{1}{2x} + \frac{1}{16} + \sin(x)$$ ### Step 5: Isolate $$p$$ To isolate $$p$$, we can move terms around: $$-p = \frac{4}{x^2} - \frac{1}{2x} + \frac{1}{16} + \sin(x) - \frac{1}{x^2} - \frac{1}{16}$$ Combine like terms: $$-p = \left(\frac{4}{x^2} - \frac{1}{x^2}\right) - \frac{1}{2x} + \sin(x)$$ $$-p = \frac{3}{x^2} - \frac{1}{2x} + \sin(x)$$ ### Step 6: Solve for $$p$$ Multiply both sides by $$-1$$ to solve for $$p$$: $$p = -\frac{3}{x^2} + \frac{1}{2x} - \sin(x)$$ ### Final Expression for $$p$$: $$p = -\frac{3}{x^2} + \frac{1}{2x} - \sin(x)$$ This is the solution for $$p$$. Do you have any questions or need further details? Here are five related questions for further exploration: 1. How can we simplify more complex expressions involving exponential functions? 2. What other types of functions can we use similar expansion techniques on? 3. How do trigonometric functions, like $$\sin(x)$$, influence the behavior of equations? 4. How do you isolate terms effectively in equations involving multiple variables? 5. What is the significance of expanding binomials in algebraic manipulations? **Tip:** When solving equations, always perform operations step-by-step and simplify each component as much as possible to avoid mistakes.