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 $C/A$.
• Maximizing a Rational Function: To maximize a fraction of the form $\frac{k}{f(x)}$, where $k$ is a positive constant, we need to minimize $f(x)$.
• Algebraic Identities:
  • $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$
  • $\alpha^3 - \beta^3 = (\alpha - \beta)(\alpha^2 + \alpha\beta + \beta^2) = (\alpha - \beta)[(\alpha + \beta)^2 - \alpha\beta]$

Step-by-Step Solution

Step 1: Identify Coefficients and Roots

The given quadratic equation is $(m^2 + 1)x^2 - 3x + (m^2 + 1)^2 = 0$.

• Why? This step is necessary to express the sum and product of the roots in terms of $m$ using Vieta's formulas.
• Comparing with the standard quadratic form $Ax^2 + Bx + C = 0$, we have:
  • $A = m^2 + 1$
  • $B = -3$
  • $C = (m^2 + 1)^2$
• Let the roots of the quadratic equation be $\alpha$ and $\beta$.

Step 2: Express Sum and Product of Roots in terms of 'm'

Using Vieta's formulas, express the sum and product of the roots in terms of $m$.

• Why? This step allows us to analyze how the sum and product of the roots change with different values of $m$.
• Sum of roots: $\alpha + \beta = -\frac{B}{A} = -\frac{-3}{m^2 + 1} = \frac{3}{m^2 + 1}$
• Product of roots: $\alpha\beta = \frac{C}{A} = \frac{(m^2 + 1)^2}{m^2 + 1} = m^2 + 1$

Step 3: Determine the Value of 'm' that Maximizes the Sum of Roots

Find the value of $m$ for which the sum of the roots, $\alpha + \beta = \frac{3}{m^2 + 1}$, is maximized.

• Why? The problem states that we need to find the value of $m$ that maximizes the sum of the roots.
• To maximize $\frac{3}{m^2 + 1}$, we need to minimize the denominator $m^2 + 1$.
• Since $m^2 \geq 0$ for all real $m$, the minimum value of $m^2$ is 0, which occurs when $m = 0$.
• Therefore, the minimum value of $m^2 + 1$ is $0 + 1 = 1$, which occurs when $m = 0$.
• Thus, the sum of the roots is greatest when $m = 0$.

Step 4: Substitute 'm' to Find Specific Sum and Product of Roots

Substitute $m = 0$ into the expressions for the sum and product of the roots.

• Why? We need to find the specific values of the sum and product of the roots for the value of $m$ that maximizes the sum of the roots.
• When $m = 0$:
  • $\alpha + \beta = \frac{3}{0^2 + 1} = 3$
  • $\alpha\beta = 0^2 + 1 = 1$

Step 5: Calculate the Absolute Difference Between the Roots

Calculate $|\alpha - \beta|$.

• Why? This is needed to calculate the absolute difference of the cubes of the roots.
• We know that $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$.
• Substituting the values $\alpha + \beta = 3$ and $\alpha\beta = 1$: $(\alpha - \beta)^2 = (3)^2 - 4(1) = 9 - 4 = 5$
• Taking the square root of both sides: $|\alpha - \beta| = \sqrt{5}$

Step 6: Calculate the Absolute Difference of the Cubes of the Roots

Calculate $|\alpha^3 - \beta^3|$.

• Why? This is the final goal of the problem.
• We have $\alpha^3 - \beta^3 = (\alpha - \beta)(\alpha^2 + \alpha\beta + \beta^2)$.
• We also know that $\alpha^2 + \alpha\beta + \beta^2 = (\alpha + \beta)^2 - \alpha\beta$.
• Therefore, $\alpha^3 - \beta^3 = (\alpha - \beta)[(\alpha + \beta)^2 - \alpha\beta]$.
• Taking the absolute value: $|\alpha^3 - \beta^3| = |\alpha - \beta||(\alpha + \beta)^2 - \alpha\beta|$
• Substituting the values $|\alpha - \beta| = \sqrt{5}$, $\alpha + \beta = 3$, and $\alpha\beta = 1$: $|\alpha^3 - \beta^3| = \sqrt{5}|(3)^2 - 1| = \sqrt{5}|9 - 1| = \sqrt{5}(8) = 8\sqrt{5}$

Common Mistakes & Tips

• Ensure you minimize the denominator when maximizing a fraction with a constant numerator.
• Remember the algebraic identities for difference of squares and cubes correctly.
• Pay attention to absolute values.

Summary

We found that the sum of the roots is maximized when $m = 0$. Substituting this value, we calculated the sum and product of the roots. Using algebraic manipulation and the values of the sum and product of the roots, we found the absolute difference of the cubes of the roots to be $8\sqrt{5}$.

The final answer is \boxed{8\sqrt{5}}, which corresponds to option (C).