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: $\alpha \beta = \frac{c}{a}$
• Geometric Progression (GP): A sequence where each term is multiplied by a constant ratio $r$ to get the next term. The terms are $a, ar, ar^2, ar^3, \dots$ where $a$ is the first term.

Step-by-Step Solution

1. Apply Vieta's Formulas to the given quadratic equations:

• For the equation $x^2 - 3x + p = 0$, with roots $\alpha$ and $\beta$:

  • Sum of roots: $\alpha + \beta = -\frac{-3}{1} = 3$
  • Product of roots: $\alpha \beta = \frac{p}{1} = p$

  Explanation: We use Vieta's formulas to relate the coefficients of the quadratic equation to the sum and product of its roots. This transforms polynomial information into algebraic equations involving the roots.

• For the equation $x^2 - 6x + q = 0$, with roots $\gamma$ and $\delta$:

  • Sum of roots: $\gamma + \delta = -\frac{-6}{1} = 6$
  • Product of roots: $\gamma \delta = \frac{q}{1} = q$

  Explanation: Same as above, but for the second quadratic equation.

2. Express the roots as terms of a Geometric Progression:

• Since $\alpha, \beta, \gamma, \delta$ form a geometric progression, let the first term be $a$ and the common ratio be $r$. Then we can represent the roots as:

  • $\alpha = a$
  • $\beta = ar$
  • $\gamma = ar^2$
  • $\delta = ar^3$

  Explanation: This step simplifies the problem by expressing the four roots in terms of only two unknowns, $a$ and $r$.

3. Substitute the GP terms into Vieta's formulas:

• From the sum of roots of the first equation: $\alpha + \beta = 3 \implies a + ar = 3 \implies a(1 + r) = 3 \quad \dots(1)$

  Explanation: Substituting the GP values for $\alpha$ and $\beta$ into the equation obtained from Vieta's formula.

• From the sum of roots of the second equation: $\gamma + \delta = 6 \implies ar^2 + ar^3 = 6 \implies ar^2(1 + r) = 6 \quad \dots(2)$

  Explanation: Substituting the GP values for $\gamma$ and $\delta$ into the equation obtained from Vieta's formula.

4. Solve for the common ratio $r$:

• Divide equation (2) by equation (1): $\frac{ar^2(1 + r)}{a(1 + r)} = \frac{6}{3}$ $r^2 = 2$ $r = \pm \sqrt{2}$

  Explanation: Dividing the two equations eliminates $a(1+r)$, leaving an equation only in terms of $r$.

5. Solve for the first term $a$:

• Substitute $r = \sqrt{2}$ into equation (1) (we can choose either root for $r$ since the final ratio will involve $r^2$ or products, and the sign will cancel out): $a(1 + \sqrt{2}) = 3$ $a = \frac{3}{1 + \sqrt{2}}$

  Explanation: Substituting the value of $r$ back into one of the original equations to solve for $a$.

6. Calculate $p$ and $q$ using $a$ and $r$:

• From the product of roots of the first equation: $p = \alpha \beta = a \cdot (ar) = a^2r$ Substitute the values of $a$ and $r$: $p = \left( \frac{3}{1 + \sqrt{2}} \right)^2 (\sqrt{2})$ $p = \frac{9\sqrt{2}}{(1 + \sqrt{2})^2} = \frac{9\sqrt{2}}{1 + 2 + 2\sqrt{2}} = \frac{9\sqrt{2}}{3 + 2\sqrt{2}}$

  Explanation: Expressing $p$ in terms of $a$ and $r$ and substituting their values.

• From the product of roots of the second equation: $q = \gamma \delta = (ar^2) \cdot (ar^3) = a^2r^5$ Substitute the values of $a$ and $r$: $q = \left( \frac{3}{1 + \sqrt{2}} \right)^2 (\sqrt{2})^5$ $q = \frac{9}{(1 + \sqrt{2})^2} \cdot 4\sqrt{2} = \frac{36\sqrt{2}}{(1 + \sqrt{2})^2} = \frac{36\sqrt{2}}{3 + 2\sqrt{2}}$

  Explanation: Expressing $q$ in terms of $a$ and $r$ and substituting their values.

7. Evaluate the required ratio $(2q + p) : (2q - p)$:

$\frac{2q + p}{2q - p} = \frac{2 \left( \frac{36\sqrt{2}}{3 + 2\sqrt{2}} \right) + \frac{9\sqrt{2}}{3 + 2\sqrt{2}}}{2 \left( \frac{36\sqrt{2}}{3 + 2\sqrt{2}} \right) - \frac{9\sqrt{2}}{3 + 2\sqrt{2}}}$

Factoring out the common term $\frac{\sqrt{2}}{3 + 2\sqrt{2}}$: $\frac{2q + p}{2q - p} = \frac{\frac{\sqrt{2}}{3 + 2\sqrt{2}}(2 \cdot 36 + 9)}{\frac{\sqrt{2}}{3 + 2\sqrt{2}}(2 \cdot 36 - 9)} = \frac{72 + 9}{72 - 9} = \frac{81}{63} = \frac{9}{7}$

Explanation: Substituting the derived values of $p$ and $q$ into the target expression and simplifying.

Common Mistakes & Tips

• Vieta's Formulas Sign: Ensure the correct signs are used when applying Vieta's formulas, especially for the sum of roots.
• Simplification: Look for common factors to simplify the expressions before final calculation.
• Sign of r: Since the final ratio involves $r^2$, choosing either $r = \sqrt{2}$ or $r = -\sqrt{2}$ will lead to the same answer. Choose the positive root for simplicity.

Summary

The problem combines quadratic equations and geometric progressions. By applying Vieta's formulas and expressing the roots as terms of a GP, we formed a system of equations. Solving for the common ratio and the first term allowed us to determine the values of $p$ and $q$, and finally, calculate the desired ratio.

The final answer is \boxed{9/7}, which corresponds to option (A).