The equation formed by multiplying each root of `ax^(2) + bx + c = 0` by 2 is `x^(2) + 36x + 24 = 0`
Which one of the following correct ?

Options

A. `bc = a^(2)`
B. `bc = 36a^(2)`
C. `bc = 72a^(2)`
D. `bc = 108a^(2)`

Answer

To solve the problem, we need to analyze the relationship between the roots of the original quadratic equation \( ax^2 + bx + c = 0 \) and the modified equation \( x^2 + 36x + 24 = 0 \).

### Step-by-Step Solution:

1. **Identify the Roots of the Original Equation**:
Let the roots of the original equation \( ax^2 + bx + c = 0 \) be \( \alpha \) and \( \beta \).

2. **Roots of the Modified Equation**:
According to the problem, when we multiply each root of the original equation by 2, the new roots become \( 2\alpha \) and \( 2\beta \). The modified equation with these roots is given as \( x^2 + 36x + 24 = 0 \).

3. **Sum of the Roots of the Modified Equation**:
The sum of the roots of the modified equation can be calculated using the formula:
\[ \text{Sum of roots} = -\frac{b}{a} \]
For the equation \( x^2 + 36x + 24 = 0 \), \( b = 36 \) and \( a = 1 \):
\[ \text{Sum of roots} = -\frac{36}{1} = -36 \]
Since the roots are \( 2\alpha + 2\beta \):
\[ 2(\alpha + \beta) = -36 \implies \alpha + \beta = -18 \]

4. **Product of the Roots of the Modified Equation**:
The product of the roots of the modified equation is given by:
\[ \text{Product of roots} = \frac{c}{a} \]
For the equation \( x^2 + 36x + 24 = 0 \), \( c = 24 \) and \( a = 1 \):
\[ \text{Product of roots} = \frac{24}{1} = 24 \]
Since the roots are \( 2\alpha \) and \( 2\beta \):
\[ (2\alpha)(2\beta) = 4\alpha\beta = 24 \implies \alpha\beta = 6 \]

5. **Relate the Sums and Products to the Original Equation**:
From Vieta's formulas for the original equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \) gives us:
\[ -\frac{b}{a} = -18 \implies b = 18a \]
- The product of the roots \( \alpha\beta = \frac{c}{a} \) gives us:
\[ \frac{c}{a} = 6 \implies c = 6a \]

6. **Calculate \( bc \)**:
Now we can find \( bc \):
\[ bc = b \cdot c = (18a)(6a) = 108a^2 \]

7. **Conclusion**:
Therefore, we have established that:
\[ bc = 108a^2 \]
This matches with option D.

### Final Answer:
The correct option is **D: \( bc = 108a^2 \)**.