Key Concepts and Formulas

• Symmetric Polynomials: Recognizing and manipulating symmetric polynomials (expressions unchanged when variables are swapped) in terms of elementary symmetric polynomials (sum and product of roots).
• Quadratic Formula: Solving quadratic equations of the form $ax^2 + bx + c = 0$ using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Discriminant: The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is $D = b^2 - 4ac$. For real roots, $D \ge 0$.

Step-by-Step Solution

Step 1: Understand the Given Information and Target Expression

We are given $p + q = 3$ and $p^4 + q^4 = 369$, where $p$ and $q$ are real numbers. Our goal is to find the value of $\left(\frac{1}{p} + \frac{1}{q}\right)^{-2}$.

Step 2: Simplify the Target Expression

First, simplify the expression we want to find: $\left(\frac{1}{p} + \frac{1}{q}\right)^{-2} = \left(\frac{p+q}{pq}\right)^{-2} = \left(\frac{pq}{p+q}\right)^{2}$ Since we know $p+q = 3$, we have: $\left(\frac{pq}{3}\right)^{2} = \frac{(pq)^2}{9}$ Thus, we need to find the value of $pq$.

Step 3: Express $p^4 + q^4$ in terms of $p+q$ and $pq$

We know that $(p+q)^2 = p^2 + 2pq + q^2$, so $p^2 + q^2 = (p+q)^2 - 2pq$. Then, $(p^2 + q^2)^2 = p^4 + 2p^2q^2 + q^4$, so $p^4 + q^4 = (p^2 + q^2)^2 - 2p^2q^2$. Substituting the expression for $p^2 + q^2$, we get: $p^4 + q^4 = \left((p+q)^2 - 2pq\right)^2 - 2(pq)^2$

Step 4: Substitute Known Values and Solve for $pq$

We are given $p+q = 3$ and $p^4 + q^4 = 369$. Substituting these values into the equation from Step 3: $369 = \left(3^2 - 2pq\right)^2 - 2(pq)^2$ $369 = (9 - 2pq)^2 - 2(pq)^2$ Let $x = pq$. Then: $369 = (9 - 2x)^2 - 2x^2$ $369 = 81 - 36x + 4x^2 - 2x^2$ $369 = 2x^2 - 36x + 81$ $0 = 2x^2 - 36x - 288$ $0 = x^2 - 18x - 144$

Step 5: Solve the Quadratic Equation for $x = pq$

Using the quadratic formula: $x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4(1)(-144)}}{2(1)}$ $x = \frac{18 \pm \sqrt{324 + 576}}{2}$ $x = \frac{18 \pm \sqrt{900}}{2}$ $x = \frac{18 \pm 30}{2}$ So, $x = \frac{18+30}{2} = \frac{48}{2} = 24$ or $x = \frac{18-30}{2} = \frac{-12}{2} = -6$. Thus, $pq = 24$ or $pq = -6$.

Step 6: Validate the Values of $pq$ using the Real Numbers Condition

Since $p$ and $q$ are real numbers, the discriminant of the quadratic $t^2 - (p+q)t + pq = 0$ must be non-negative. The discriminant is $(p+q)^2 - 4pq \ge 0$. Substituting $p+q = 3$, we have $3^2 - 4pq \ge 0$, so $9 - 4pq \ge 0$, which means $4pq \le 9$, or $pq \le \frac{9}{4} = 2.25$. Since $24 > 2.25$ and $-6 < 2.25$, we must have $pq = -6$.

Step 7: Calculate the Final Expression

We want to find $\frac{(pq)^2}{9}$. Since $pq = -6$, we have: $\frac{(pq)^2}{9} = \frac{(-6)^2}{9} = \frac{36}{9} = 4$

Common Mistakes & Tips

• Forgetting the Real Root Condition: Always check if the solutions obtained satisfy the condition for real roots using the discriminant.
• Sign Errors: Be careful with signs when applying the quadratic formula and simplifying expressions.
• Simplifying Early: Simplify the target expression as much as possible before substituting values to reduce complexity.

Summary

By expressing $p^4 + q^4$ in terms of $p+q$ and $pq$, we formed a quadratic equation for $pq$. Solving this equation gave two possible values for $pq$, but only one satisfied the condition for $p$ and $q$ to be real numbers. Substituting this value into the simplified target expression gave the final answer.

The final answer is $\boxed{4}$.