Key Concepts and Formulas

• Vieta's Formulas: For a quadratic equation $Ax^2 + Bx + C = 0$, the sum of the roots is $-B/A$ and the product of the roots is $C/A$.
• Repeated Roots: A quadratic equation $Ax^2 + Bx + C = 0$ has a repeated root if and only if its discriminant is zero, i.e., $B^2 - 4AC = 0$. Also, the repeated root is given by $\alpha = -B/(2A)$.
• Common Roots: If a value is a root of an equation, substituting it into the equation will satisfy the equation.

Step-by-Step Solution

Step 1: Analyzing the First Equation ($ax^2 - 2bx + 5 = 0$)

The problem states that the equation $ax^2 - 2bx + 5 = 0$ has a repeated root, denoted by $\alpha$. Since $a \ne 0$, this is a standard quadratic equation. We use Vieta's formulas and the repeated root condition to establish relationships between $\alpha$, $a$, and $b$.

• Using Vieta's Formulas for Repeated Roots:
  • Sum of roots: $\alpha + \alpha = 2\alpha$. According to Vieta's formulas, this sum is equal to $-\frac{-2b}{a} = \frac{2b}{a}$. $2\alpha = \frac{2b}{a}$ Simplifying this, we get: $\alpha = \frac{b}{a} \quad (*)$
    • Explanation: This step uses the sum of roots property to establish a relationship between the root $\alpha$ and the coefficients $a$ and $b$.
  • Product of roots: $\alpha \cdot \alpha = \alpha^2$. According to Vieta's formulas, this product is equal to $\frac{5}{a}$. $\alpha^2 = \frac{5}{a} \quad (**)$
    • Explanation: This step uses the product of roots property to establish another relationship involving $\alpha$, $a$, and $5$.

Step 2: Analyzing the Second Equation ($x^2 - 2bx - 10 = 0$)

The problem states that $\alpha$ is also a root of the equation $x^2 - 2bx - 10 = 0$. Let the other root be $\beta$. We use Vieta's formulas for this equation to establish relationships between $\alpha$, $\beta$, and $b$.

• Using Vieta's Formulas for the Second Equation:
  • Sum of roots: $\alpha + \beta = -\frac{-2b}{1} = 2b$. $\alpha + \beta = 2b \quad (1)$
    • Explanation: This relates the two roots of the second equation to the coefficient $b$.
  • Product of roots: $\alpha \beta = \frac{-10}{1} = -10$. $\alpha \beta = -10 \quad (2)$
    • Explanation: This provides a direct relationship between the two roots.

Step 3: Using the Common Root Condition

Since $\alpha$ is a root of $x^2 - 2bx - 10 = 0$, it must satisfy the equation. We substitute $\alpha$ into the equation to obtain a further relationship.

• Substituting $\alpha$ into the Second Equation: $\alpha^2 - 2b\alpha - 10 = 0 \quad (3)$
  • Explanation: This is a fundamental property: if a value is a root of an equation, it makes the equation true when substituted.

Step 4: Solving for Coefficients and Roots

Now we combine the information from the two equations to find the values of $a$, $\alpha^2$, and $\beta^2$.

• Relating $a$, $b$, and $\alpha$: From equation $(*)$, we have $b = a\alpha$.

  • Explanation: We rearrange the relation found in Step 1 to express $b$ in terms of $a$ and $\alpha$.

• Substituting $b$ into the Common Root Equation: Substitute $b = a\alpha$ into the equation $\alpha^2 - 2b\alpha - 10 = 0$: $\alpha^2 - 2(a\alpha)\alpha - 10 = 0$ $\alpha^2 - 2a\alpha^2 - 10 = 0$ $\alpha^2(1 - 2a) = 10 \quad (4)$

  • Explanation: We eliminate $b$ from the equation by using its relationship with $a$ and $\alpha$, creating an equation solely in terms of $a$ and $\alpha$.

• Solving for $a$: Now, substitute $\alpha^2 = \frac{5}{a}$ (from equation $(**)$) into $\alpha^2(1 - 2a) = 10$: $\left(\frac{5}{a}\right)(1 - 2a) = 10$ $5(1 - 2a) = 10a \quad (\text{Multiply both sides by } a)$ $5 - 10a = 10a$ $5 = 20a$ $a = \frac{5}{20} = \frac{1}{4}$

  • Explanation: This yields the numerical value of the coefficient $a$.

• Finding $\alpha^2$: Now that we have $a$, we can find $\alpha^2$ using equation $(**)$: $\alpha^2 = \frac{5}{a} = \frac{5}{1/4} = 5 \times 4 = 20$

  • Explanation: We've found the value of $\alpha^2$.

• Finding $\beta$: From equation (1), $\alpha + \beta = 2b$. Substituting $b=a\alpha$ gives $\alpha + \beta = 2a\alpha$. Substituting $a = \frac{1}{4}$ gives $\alpha + \beta = \frac{1}{2}\alpha$. Therefore, $\beta = \frac{1}{2}\alpha - \alpha = -\frac{1}{2}\alpha$.

• Finding $\beta^2$: $\beta^2 = \left(-\frac{1}{2}\alpha\right)^2 = \frac{1}{4}\alpha^2 = \frac{1}{4}(20) = 5$

  • Explanation: We've found the value of $\beta^2$.

Step 5: Calculating $\alpha^2 + \beta^2$

We now have the values for $\alpha^2$ and $\beta^2$.

• Sum of Squares: $\alpha^2 + \beta^2 = 20 + 5 = 25$
  • Explanation: We simply add the calculated values of $\alpha^2$ and $\beta^2$ to get the final answer.

Common Mistakes & Tips

• Algebraic Errors: Making mistakes in algebraic manipulations, especially when substituting expressions or solving for variables like $a$ or $b$. Double-check each step.
• Confusing Equations: Clearly label equations and the source of each equation to prevent mixing up relationships.
• Not using all information: Make sure to use ALL conditions given in the problem, like repeated roots and common roots.

Summary

This problem requires a solid understanding of quadratic equations, Vieta's formulas, and the concept of repeated and common roots. By systematically applying these concepts, we derived a system of equations that allowed us to solve for the unknown values of $a$, $\alpha^2$, and $\beta^2$. Finally, we calculated $\alpha^2 + \beta^2 = 25$.

The final answer is \boxed{25}, which corresponds to option (D).