If m is chosen in the quadratic equation `(m^(2)+1)x^(2)-3x+(m^(2)+1)^(2)=0` such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is

Options

A. `8 sqrt(5)`
B. `4 sqrt(3)`
C. `8 sqrt(3)`
D. `10 sqrt(5)`

Explanation

To solve the problem, we need to analyze the quadratic equation given and find the absolute difference of the cubes of its roots when the sum of the roots is maximized.

### Step-by-step Solution:

1. **Write the quadratic equation**:
The quadratic equation is given as:
\[ (m^2 + 1)x^2 - 3x + (m^2 + 1)^2 = 0 \]

2. **Divide by \(m^2 + 1\)**:
To simplify, we can divide the entire equation by \(m^2 + 1\) (assuming \(m^2 + 1 \neq 0\)):
\[ x^2 - \frac{3}{m^2 + 1}x + (m^2 + 1) = 0 \]

3. **Find the sum of the roots**:
The sum of the roots \(S\) of the quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[ S = -\frac{b}{a} \]
Here, \(a = 1\) and \(b = -\frac{3}{m^2 + 1}\). Thus, the sum of the roots is:
\[ S = \frac{3}{m^2 + 1} \]

4. **Maximize the sum of the roots**:
To maximize \(S\), we need to minimize \(m^2 + 1\). The minimum value of \(m^2 + 1\) occurs when \(m = 0\):
\[ m^2 + 1 = 1 \quad \text{(when \(m = 0\))} \]
Therefore, the maximum sum of the roots is:
\[ S = 3 \]

5. **Substitute \(m = 0\) back into the equation**:
The quadratic equation becomes:
\[ x^2 - 3x + 1 = 0 \]

6. **Find the roots using the quadratic formula**:
The roots \(\alpha\) and \(\beta\) can be found using:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -3\), and \(c = 1\):
\[ x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{3 \pm \sqrt{9 - 4}}{2} = \frac{3 \pm \sqrt{5}}{2} \]
Thus, the roots are:
\[ \alpha = \frac{3 + \sqrt{5}}{2}, \quad \beta = \frac{3 - \sqrt{5}}{2} \]

7. **Calculate the absolute difference of the cubes of the roots**:
We need to find \(|\alpha^3 - \beta^3|\). Using the identity for the difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, \(a = \alpha\) and \(b = \beta\):
\[ \alpha - \beta = \frac{3 + \sqrt{5}}{2} - \frac{3 - \sqrt{5}}{2} = \sqrt{5} \]

8. **Calculate \(a^2 + ab + b^2\)**:
We can find \(a^2 + b^2\) using:
\[ a^2 + b^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
We have:
\[ \alpha + \beta = 3, \quad \alpha \beta = 1 \]
Thus,
\[ a^2 + b^2 = 3^2 - 2 \cdot 1 = 9 - 2 = 7 \]
Therefore,
\[ a^2 + ab + b^2 = 7 + 1 = 8 \]

9. **Combine results**:
Now substituting back:
\[ |\alpha^3 - \beta^3| = |\alpha - \beta| \cdot (\alpha^2 + \alpha\beta + \beta^2) = \sqrt{5} \cdot 8 = 8\sqrt{5} \]

### Final Answer:
The absolute difference of the cubes of its roots is:
\[ \boxed{8\sqrt{5}} \]