The equation whose roots are double the roots of the equation `x^(2) + 6x + 3 = 0` is

Options

A. `2x^(2) + 12x + 6 =0`
B. `4x^(2) + 12x + 3 =0`
C. `x^(2) + 12x + 12 =0`
D. `2x^(2) + 12x + 12 =0`

Answer

To find the equation whose roots are double the roots of the given equation \(x^2 + 6x + 3 = 0\), we can follow these steps:

### Step 1: Identify the roots of the given equation
The given equation is:
\[ x^2 + 6x + 3 = 0 \]
We can use the formulas for the sum and product of the roots of a quadratic equation \(ax^2 + bx + c = 0\):
- Sum of roots (\(\alpha + \beta\)) = \(-\frac{b}{a}\)
- Product of roots (\(\alpha \beta\)) = \(\frac{c}{a}\)

Here, \(a = 1\), \(b = 6\), and \(c = 3\).

Calculating the sum and product:
- \(\alpha + \beta = -\frac{6}{1} = -6\)
- \(\alpha \beta = \frac{3}{1} = 3\)

### Step 2: Determine the new roots
Since we need to find the equation whose roots are double the roots of the original equation, the new roots will be:
\[ 2\alpha \quad \text{and} \quad 2\beta \]

### Step 3: Calculate the sum and product of the new roots
The sum of the new roots:
\[ 2\alpha + 2\beta = 2(\alpha + \beta) = 2(-6) = -12 \]

The product of the new roots:
\[ (2\alpha)(2\beta) = 4(\alpha \beta) = 4 \times 3 = 12 \]

### Step 4: Form the new quadratic equation
Using the sum and product of the new roots, we can form the new quadratic equation in the standard form:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - (-12)x + 12 = 0 \]
This simplifies to:
\[ x^2 + 12x + 12 = 0 \]

### Final Answer
Thus, the equation whose roots are double the roots of the equation \(x^2 + 6x + 3 = 0\) is:
\[ \boxed{x^2 + 12x + 12 = 0} \]