Key Concepts and Formulas

• Roots of a Quadratic Equation: If $r$ is a root of the quadratic equation $ax^2+bx+c=0$, then $r$ satisfies the equation, i.e., $ar^2+br+c=0$.
• Sum of Powers of Roots: The expressions $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$ can be simplified by using the relationships derived from the quadratic equations.
• Algebraic Manipulation: Factoring, substitution, and simplification are crucial techniques for solving this problem.

Step-by-Step Solution

Part 1: Evaluating $\frac{P_{25}+\sqrt{3} P_{24}}{2 P_{23}}$

Step 1: Analyze the first quadratic equation. The quadratic equation is $x^2 + \sqrt{3}x - 16 = 0$, with roots $\alpha$ and $\beta$. Therefore, $\alpha^2 + \sqrt{3}\alpha - 16 = 0$ and $\beta^2 + \sqrt{3}\beta - 16 = 0$. We rearrange to get $\alpha^2 + \sqrt{3}\alpha = 16$ and $\beta^2 + \sqrt{3}\beta = 16$. Reasoning: This step establishes the fundamental relationships that the roots must satisfy.

Step 2: Expand the numerator using the definition of $P_n$. We have $P_{25} + \sqrt{3}P_{24} = (\alpha^{25} + \beta^{25}) + \sqrt{3}(\alpha^{24} + \beta^{24}) = \alpha^{25} + \sqrt{3}\alpha^{24} + \beta^{25} + \sqrt{3}\beta^{24}$. Reasoning: This step expresses the numerator in terms of the roots $\alpha$ and $\beta$.

Step 3: Factor and substitute using the relationships from Step 1. Factoring gives $\alpha^{23}(\alpha^2 + \sqrt{3}\alpha) + \beta^{23}(\beta^2 + \sqrt{3}\beta)$. Substituting $\alpha^2 + \sqrt{3}\alpha = 16$ and $\beta^2 + \sqrt{3}\beta = 16$, we get $\alpha^{23}(16) + \beta^{23}(16) = 16(\alpha^{23} + \beta^{23}) = 16P_{23}$. Reasoning: This step utilizes the relationships derived in Step 1 to simplify the expression.

Step 4: Simplify the first expression. We have $\frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} = \frac{16P_{23}}{2P_{23}} = 8$. We assume $P_{23} \neq 0$, which is valid since the roots are real and non-zero. Reasoning: This step simplifies the expression to a numerical value.

Part 2: Evaluating $\frac{Q_{25}-Q_{23}}{Q_{24}}$

Step 5: Analyze the second quadratic equation. The quadratic equation is $x^2 + 3x - 1 = 0$, with roots $\gamma$ and $\delta$. Therefore, $\gamma^2 + 3\gamma - 1 = 0$ and $\delta^2 + 3\delta - 1 = 0$. We rearrange to get $\gamma^2 - 1 = -3\gamma$ and $\delta^2 - 1 = -3\delta$. Reasoning: This step establishes the fundamental relationships that the roots must satisfy.

Step 6: Expand the numerator using the definition of $Q_n$. We have $Q_{25} - Q_{23} = (\gamma^{25} + \delta^{25}) - (\gamma^{23} + \delta^{23}) = \gamma^{25} - \gamma^{23} + \delta^{25} - \delta^{23}$. Reasoning: This step expresses the numerator in terms of the roots $\gamma$ and $\delta$.

Step 7: Factor and substitute using the relationships from Step 5. Factoring gives $\gamma^{23}(\gamma^2 - 1) + \delta^{23}(\delta^2 - 1)$. Substituting $\gamma^2 - 1 = -3\gamma$ and $\delta^2 - 1 = -3\delta$, we get $\gamma^{23}(-3\gamma) + \delta^{23}(-3\delta) = -3\gamma^{24} - 3\delta^{24} = -3(\gamma^{24} + \delta^{24}) = -3Q_{24}$. Reasoning: This step utilizes the relationships derived in Step 5 to simplify the expression.

Step 8: Simplify the second expression. We have $\frac{Q_{25} - Q_{23}}{Q_{24}} = \frac{-3Q_{24}}{Q_{24}} = -3$. We assume $Q_{24} \neq 0$, which is valid since the roots are real and non-zero and 24 is even, making $Q_{24}$ positive. Reasoning: This step simplifies the expression to a numerical value.

Step 9: Combine the results. The original question asks for the value of $\frac{P_{25}+\sqrt{3} P_{24}}{2 P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}}$. Substituting the values we found, we get $8 + (-3) = 5$. Reasoning: This step combines the results from the previous steps to find the final answer.

Common Mistakes & Tips

• Be careful with signs when substituting and simplifying.
• Always check for potential division by zero before cancelling terms.
• Utilize the quadratic equation relationships to simplify the expressions involving powers of roots.

Summary

By leveraging the properties of quadratic equations and their roots, we simplified the given expression step by step. We first found the values of $\frac{P_{25}+\sqrt{3} P_{24}}{2 P_{23}}$ and $\frac{Q_{25}-Q_{23}}{Q_{24}}$ separately and then added them to obtain the final result. The final answer is 5.

Final Answer The final answer is \boxed{5}, which corresponds to option (C).