If the equation `x^(3) + ax^(2) + b = 0, b ne 0` has a root of order 2, then

Options

A. `a^(2) + 2b = 0`
B. `a^(2) - 2b = 0`
C. `4a^(3) + 27b + 1 = 0`
D. `4a^(3) + 27b = 0`

Answer

To solve the problem, we need to analyze the cubic equation given and the conditions about its roots.

### Step-by-Step Solution:

1. **Understanding the Roots**:
We are given the cubic equation \( x^3 + ax^2 + b = 0 \) (where \( b \neq 0 \)) has a root of order 2. This means that one of the roots, say \( \alpha \), is repeated. Therefore, the roots can be expressed as \( \alpha, \alpha, \beta \).

2. **Sum of the Roots**:
According to Vieta's formulas, the sum of the roots of the polynomial \( x^3 + ax^2 + b = 0 \) is equal to \( -a \). Thus, we can write:
\[ \alpha + \alpha + \beta = -a \]
Simplifying this gives:
\[ 2\alpha + \beta = -a \quad \text{(Equation 1)} \]

3. **Sum of the Product of Roots Taken Two at a Time**:
The sum of the products of the roots taken two at a time is equal to 0 (since the coefficient of \( x \) is 0). Therefore:
\[ \alpha^2 + \alpha\beta + \alpha\beta = 0 \]
This simplifies to:
\[ \alpha^2 + 2\alpha\beta = 0 \]
Factoring out \( \alpha \):
\[ \alpha(\alpha + 2\beta) = 0 \]
This gives us two cases:
- Case 1: \( \alpha = 0 \)
- Case 2: \( \alpha + 2\beta = 0 \) which implies \( \alpha = -2\beta \)

4. **Product of the Roots**:
The product of the roots is given by:
\[ \alpha^2 \beta = -b \]

5. **Analyzing Case 1**:
If \( \alpha = 0 \):
\[ 0^2 \cdot \beta = -b \implies 0 = -b \]
This contradicts the condition \( b \neq 0 \). Thus, Case 1 is not valid.

6. **Analyzing Case 2**:
If \( \alpha = -2\beta \), we substitute this into Equation 1:
\[ 2(-2\beta) + \beta = -a \]
Simplifying gives:
\[ -4\beta + \beta = -a \implies -3\beta = -a \implies \beta = \frac{a}{3} \]
Now substituting \( \beta \) back to find \( \alpha \):
\[ \alpha = -2\beta = -2 \cdot \frac{a}{3} = -\frac{2a}{3} \]

7. **Finding the Product of the Roots**:
Now substituting \( \alpha \) and \( \beta \) into the product of roots equation:
\[ \left(-\frac{2a}{3}\right)^2 \cdot \frac{a}{3} = -b \]
This simplifies to:
\[ \frac{4a^2}{9} \cdot \frac{a}{3} = -b \implies \frac{4a^3}{27} = -b \]
Rearranging gives:
\[ 4a^3 + 27b = 0 \]

### Final Result:
Thus, the relationship we derived is:
\[ 4a^3 + 27b = 0 \]