Key Concepts and Formulas

• Vieta's Formulas: For a quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$:
  • Sum of roots: $\alpha + \beta = -\frac{b}{a}$
  • Product of roots: $\alpha \beta = \frac{c}{a}$
• Tangent Addition Formula: $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

Step-by-Step Solution

Step 1: Apply Vieta's Formulas

Given the quadratic equation $x^2 + px + q = 0$ with roots $\tan 30^\circ$ and $\tan 15^\circ$. By Vieta's formulas:

• Sum of roots: $\tan 30^\circ + \tan 15^\circ = -p$. This is because the sum of the roots is the negative of the coefficient of $x$ divided by the coefficient of $x^2$, which is $-p/1 = -p$.

$\tan 30^\circ + \tan 15^\circ = -p \quad \text{(Equation 1)}$

• Product of roots: $\tan 30^\circ \cdot \tan 15^\circ = q$. This is because the product of the roots is the constant term divided by the coefficient of $x^2$, which is $q/1 = q$.

$\tan 30^\circ \cdot \tan 15^\circ = q \quad \text{(Equation 2)}$

Step 2: Use the Tangent Addition Formula

Consider the tangent addition formula with $A = 30^\circ$ and $B = 15^\circ$. Then $A + B = 45^\circ$, and we know that $\tan 45^\circ = 1$.

Substitute $A = 30^\circ$ and $B = 15^\circ$ into the tangent addition formula:

$\tan(30^\circ + 15^\circ) = \frac{\tan 30^\circ + \tan 15^\circ}{1 - \tan 30^\circ \tan 15^\circ}$

Since $\tan(30^\circ + 15^\circ) = \tan 45^\circ = 1$, we have:

$1 = \frac{\tan 30^\circ + \tan 15^\circ}{1 - \tan 30^\circ \tan 15^\circ}$

This step is crucial because it relates the sum and product of the roots to a known value.

Step 3: Substitute from Vieta's Formulas

Substitute the values from Equations 1 and 2 into the tangent addition formula:

$1 = \frac{-p}{1 - q}$

This substitution allows us to express the given condition in terms of $p$ and $q$.

Step 4: Solve for $q - p$

Solve the equation for $q - p$:

$1 = \frac{-p}{1 - q}$ $1 - q = -p$ $1 = q - p$ $q - p = 1 \quad \text{(Equation 3)}$

This algebraic manipulation allows us to isolate the term $q-p$.

Step 5: Calculate the Value of $2 + q - p$

The problem asks for the value of $2 + q - p$. Substitute the value of $q - p$ from Equation 3:

$2 + q - p = 2 + 1 = 3$

This is the final calculation to find the answer.

Common Mistakes & Tips

• Avoid Direct Calculation: Do not calculate $\tan 30^\circ$ and $\tan 15^\circ$ individually and then substitute. This approach is more complex and prone to errors.
• Recognize the Tangent Addition Formula: The key to solving this problem efficiently is recognizing the applicability of the tangent addition formula.
• Master Basic Trigonometric Values: Knowing the trigonometric values of common angles (e.g., $30^\circ$, $45^\circ$, $60^\circ$) is essential.

Summary

The problem combines Vieta's formulas with the tangent addition formula. Recognizing that $\tan(30^\circ + 15^\circ) = \tan 45^\circ = 1$ is crucial for simplifying the problem. By substituting the sum and product of the roots from Vieta's formulas into the tangent addition formula, we find that $q - p = 1$. Therefore, $2 + q - p = 2 + 1 = 3$.

Final Answer

The final answer is \boxed{3}, which corresponds to option (B).