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

Options

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

Explanation

To solve the problem, we need to find the ratio \( \frac{b}{c} \) given the equation formed by multiplying each root of \( ax^2 + bx + c = 0 \) by 2, which results in the equation \( x^2 + 36x + 24 = 0 \).

### Step-by-Step Solution:

1. **Identify the Roots of the New Equation:**
The new equation is \( x^2 + 36x + 24 = 0 \). The roots of this equation can be denoted as \( 2\alpha \) and \( 2\beta \), where \( \alpha \) and \( \beta \) are the roots of the original equation \( ax^2 + bx + c = 0 \).

2. **Sum of the Roots:**
The sum of the roots of the new equation is given by:
\[ 2\alpha + 2\beta = -\text{(coefficient of } x) = -36 \]
Dividing both sides by 2 gives:
\[ \alpha + \beta = -18 \]

3. **Product of the Roots:**
The product of the roots of the new equation is given by:
\[ (2\alpha)(2\beta) = 4\alpha\beta = \text{(constant term)} = 24 \]
Dividing both sides by 4 gives:
\[ \alpha\beta = 6 \]

4. **Relate to Original Equation:**
From Vieta's formulas for the original equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha\beta = \frac{c}{a} \)

From our previous results:
\[ -\frac{b}{a} = -18 \quad \text{(1)} \]
\[ \frac{c}{a} = 6 \quad \text{(2)} \]

5. **Solve for \( b \) and \( c \):**
From equation (1):
\[ b = 18a \]
From equation (2):
\[ c = 6a \]

6. **Find the Ratio \( \frac{b}{c} \):**
Now, we can find the ratio \( \frac{b}{c} \):
\[ \frac{b}{c} = \frac{18a}{6a} = \frac{18}{6} = 3 \]

Thus, the ratio \( \frac{b}{c} \) is \( 3:1 \).

### Final Answer:
\[ \frac{b}{c} = 3:1 \]