Key Concepts and Formulas

• Algebraic Manipulation: Expanding and simplifying polynomial expressions.
• Substitution: Using a new variable to simplify complex equations.
• Solving Quadratic Equations: Finding roots by factoring or using the quadratic formula ($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$).
• Nature of Roots: Determining if roots are rational or irrational based on the discriminant ($\Delta = b^2 - 4ac$). Rational roots occur when $\Delta$ is a perfect square.

Step-by-Step Solution

Step 1: Simplify the Equation

We are given the equation: $(x^2 - 9x + 11)^2 - (x-4)(x-5) = 3$ First, we expand the product $(x-4)(x-5)$: $(x-4)(x-5) = x^2 - 5x - 4x + 20 = x^2 - 9x + 20$ Substituting this back into the original equation gives: $(x^2 - 9x + 11)^2 - (x^2 - 9x + 20) = 3$ Why this step? Expanding the product helps reveal a common algebraic expression that allows for simplification using substitution.

Step 2: Introduce Substitution

Notice that the expression $x^2 - 9x$ appears in multiple terms. Let's substitute: $t = x^2 - 9x$ Then we can rewrite the terms in the equation as: $x^2 - 9x + 11 = t + 11$ $x^2 - 9x + 20 = t + 20$ Why this step? This substitution transforms the original quartic equation into a quadratic equation in terms of $t$, making it easier to solve.

Step 3: Solve for the Intermediate Variable ($t$)

Substitute these expressions back into the equation: $(t+11)^2 - (t+20) = 3$ Expand and simplify: $t^2 + 22t + 121 - t - 20 = 3$ Combine like terms: $t^2 + 21t + 101 = 3$ Move the constant term to the left side: $t^2 + 21t + 98 = 0$ Why this step? We've converted the problem into solving a standard quadratic equation for $t$.

Step 4: Factor the Quadratic in $t$ and Solve for $t$

We need to find two numbers that multiply to 98 and add up to 21. These numbers are 7 and 14. Therefore, we can factor the quadratic equation as: $(t+7)(t+14) = 0$ This gives us two possible values for $t$: $t+7 = 0 \implies t = -7$ $t+14 = 0 \implies t = -14$ Why this step? Factoring allows us to find the specific values of $t$ that satisfy the equation, which are necessary for finding the roots of $x$.

Step 5: Substitute Back to Find Roots of $x$

Now, we substitute $t = x^2 - 9x$ back into the equation for each value of $t$ to find the roots for $x$.

• Case 1: $t = -7$ Substitute $t$ back: $x^2 - 9x = -7$ Rearrange into the standard quadratic form $ax^2 + bx + c = 0$: $x^2 - 9x + 7 = 0$ Why this step? We are now solving for the roots of $x$ corresponding to one of the values of $t$. The discriminant is $\Delta = b^2 - 4ac = (-9)^2 - 4(1)(7) = 81 - 28 = 53$. Since $\Delta = 53$ is not a perfect square, the roots $x = \frac{9 \pm \sqrt{53}}{2}$ are irrational. As the question asks for the product of rational roots, these are excluded.

• Case 2: $t = -14$ Substitute $t$ back: $x^2 - 9x = -14$ Rearrange into the standard quadratic form: $x^2 - 9x + 14 = 0$ Why this step? We solve for the roots of $x$ for the second value of $t$. We look for two numbers that multiply to 14 and add up to -9, which are -2 and -7. $(x-2)(x-7) = 0$ The roots are: $x - 2 = 0 \implies x = 2$ $x - 7 = 0 \implies x = 7$ These roots, $x=2$ and $x=7$, are integers and are therefore rational roots.

Step 6: Identify Rational Roots and Calculate Their Product

The rational roots identified from both cases are $x=2$ and $x=7$. The question asks for the product of all the rational roots. Product of rational roots $= 2 \times 7 = 14$.

Common Mistakes & Tips

• Recognize Patterns for Substitution: Always look for repeating expressions or symmetries that suggest substitution.
• Understand Root Classifications: Be clear about the difference between rational and irrational roots. Use the discriminant to check if roots are rational without solving.
• Verify the Question's Requirements: Pay close attention to what the question asks for, ensuring you are calculating the correct quantity (product, sum) and including only the specified type of roots (rational, real).

Summary

By simplifying the given equation through algebraic manipulation and substitution, we transformed it into a solvable quadratic equation. We found two values for $t$, which led to two quadratic equations for $x$. One quadratic resulted in irrational roots, while the other yielded rational roots $x=2$ and $x=7$. The product of these rational roots is $2 \times 7 = 14$.

Final Answer

The final answer is $\boxed{14}$, which corresponds to option (D).