The equation formed by multiplying each root of `ax^(2) + bx + c = 0` by 2 is `x^(2) + 36x + 24 = 0`
What is the value of b : c ?

Options

A. `3 : 1`
B. `1 : 2`
C. `1 : 3`
D. `3 : 2`

Answer

To solve the problem, we need to find the ratio \( b : c \) given that the equation formed by multiplying each root of \( ax^2 + bx + c = 0 \) by 2 is \( x^2 + 36x + 24 = 0 \).

### Step-by-Step Solution:

1. **Identify the roots of the second equation**:
The equation given is \( x^2 + 36x + 24 = 0 \). We can denote the roots of this equation as \( 2\alpha \) and \( 2\beta \), where \( \alpha \) and \( \beta \) are the roots of the original equation \( ax^2 + bx + c = 0 \).

2. **Use Vieta's formulas**:
According to Vieta's formulas:
- The sum of the roots \( (2\alpha + 2\beta) \) of the second equation is equal to \( -\frac{b}{a} \).
- The product of the roots \( (2\alpha \cdot 2\beta) \) is equal to \( \frac{c}{a} \).

3. **Calculate the sum of the roots**:
From the second equation:
\[ 2\alpha + 2\beta = -36 \]
Dividing by 2:
\[ \alpha + \beta = -18 \]
According to Vieta's for the first equation, this gives us:
\[ -\frac{b}{a} = -18 \implies b = 18a \]

4. **Calculate the product of the roots**:
From the second equation:
\[ 2\alpha \cdot 2\beta = 24 \implies 4\alpha\beta = 24 \]
Dividing by 4:
\[ \alpha\beta = 6 \]
According to Vieta's for the first equation, this gives us:
\[ \frac{c}{a} = 6 \implies c = 6a \]

5. **Find the ratio \( b : c \)**:
Now we have:
\[ b = 18a \quad \text{and} \quad c = 6a \]
Thus, the ratio \( b : c \) is:
\[ \frac{b}{c} = \frac{18a}{6a} = \frac{18}{6} = 3 \]
Therefore, the ratio \( b : c = 3 : 1 \).

### Final Answer:
The value of \( b : c \) is \( 3 : 1 \).