Key Concepts and Formulas

• Conjugate Root Theorem: If a polynomial equation with rational coefficients has a root of the form $a + \sqrt{b}$, where $a$ and $b$ are rational and $\sqrt{b}$ is irrational, then $a - \sqrt{b}$ is also a root.
• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is given by $-\frac{b}{a}$, and the product of the roots is given by $\frac{c}{a}$.

Step-by-Step Solution

Step 1: Identify the Roots

• We are given that one root of the quadratic equation $x^2 + px + q = 0$ is $x_1 = 2 - \sqrt{3}$.
• Since $p, q \in \mathbb{R}$, we assume $p$ and $q$ are rational numbers. Therefore, we can apply the Conjugate Root Theorem.
• The Conjugate Root Theorem states that if $2 - \sqrt{3}$ is a root, then its conjugate $2 + \sqrt{3}$ is also a root.
• Thus, the second root is $x_2 = 2 + \sqrt{3}$.
• Why: This step utilizes the Conjugate Root Theorem, a fundamental concept for quadratic equations with rational coefficients, to determine the second root given one irrational root.

Step 2: Apply Vieta's Formulas

• For the quadratic equation $x^2 + px + q = 0$, Vieta's formulas relate the sum and product of the roots to the coefficients $p$ and $q$.
• The sum of the roots is $x_1 + x_2 = -p$.
• The product of the roots is $x_1 \cdot x_2 = q$.
• Why: Vieta's formulas provide a direct link between the roots of the quadratic and its coefficients, simplifying the process of finding $p$ and $q$.

Step 3: Calculate the Sum of the Roots

• We have $x_1 = 2 - \sqrt{3}$ and $x_2 = 2 + \sqrt{3}$.
• Therefore, $x_1 + x_2 = (2 - \sqrt{3}) + (2 + \sqrt{3}) = 2 - \sqrt{3} + 2 + \sqrt{3} = 4$.
• Since $x_1 + x_2 = -p$, we have $-p = 4$, so $p = -4$.
• Why: Calculating the sum of the roots allows us to directly determine the value of $p$ using Vieta's formulas.

Step 4: Calculate the Product of the Roots

• We have $x_1 = 2 - \sqrt{3}$ and $x_2 = 2 + \sqrt{3}$.
• Therefore, $x_1 \cdot x_2 = (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1$.
• Since $x_1 \cdot x_2 = q$, we have $q = 1$.
• Why: Calculating the product of the roots allows us to directly determine the value of $q$ using Vieta's formulas.

Step 5: Verify the Options

• Now that we have found $p = -4$ and $q = 1$, we substitute these values into each option to find the correct relationship.

  • (A) $p^2 - 4q - 12 = 0$

    • $(-4)^2 - 4(1) - 12 = 16 - 4 - 12 = 0$. This is correct.

  • (B) $q^2 - 4p - 16 = 0$

    • $(1)^2 - 4(-4) - 16 = 1 + 16 - 16 = 1 \neq 0$.

  • (C) $q^2 + 4p + 14 = 0$

    • $(1)^2 + 4(-4) + 14 = 1 - 16 + 14 = -1 \neq 0$.

  • (D) $p^2 - 4q + 12 = 0$

    • $(-4)^2 - 4(1) + 12 = 16 - 4 + 12 = 24 \neq 0$.

• Only option (A) holds true when $p = -4$ and $q = 1$.

• Why: This step systematically checks each option to identify the one that is satisfied by the calculated values of $p$ and $q$, ensuring we arrive at the correct answer.

Common Mistakes & Tips

• Remember to check if the coefficients are rational before applying the Conjugate Root Theorem. If not, this theorem is not applicable.
• Pay close attention to the signs in Vieta's formulas. The sum of the roots is $-b/a$, not $b/a$.
• Be careful when expanding expressions, especially when dealing with square roots.

Summary By using the Conjugate Root Theorem to find the second root and Vieta's formulas to relate the roots to the coefficients, we determined that $p = -4$ and $q = 1$. Substituting these values into the given options, we found that the correct relationship between $p$ and $q$ is $p^2 - 4q - 12 = 0$, which corresponds to option (A).

Final Answer The final answer is \boxed{p^2 – 4q – 12 = 0}, which corresponds to option (A).