If `(1)/(2-sqrt(-2))` is one of the root of `ax^(2) + bx + c = 0`. Where a,b and c are real, then what are the value of a.b.c respectively.

Options

A. 6, -4,1
B. 4,6,-1
C. 3,-2,1
D. 6,4,1

Answer

To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) in the quadratic equation \( ax^2 + bx + c = 0 \) given that one of the roots is \( \frac{1}{2 - \sqrt{-2}} \).

### Step 1: Simplify the root
The root given is \( \frac{1}{2 - \sqrt{-2}} \). We can rewrite \( \sqrt{-2} \) as \( i\sqrt{2} \) (where \( i \) is the imaginary unit).

So, we have:
\[ \frac{1}{2 - i\sqrt{2}} \]

### Step 2: Multiply by the conjugate
To simplify this expression, we multiply the numerator and the denominator by the conjugate of the denominator:
\[ \frac{1 \cdot (2 + i\sqrt{2})}{(2 - i\sqrt{2})(2 + i\sqrt{2})} \]

### Step 3: Calculate the denominator
The denominator simplifies as follows:
\[ (2 - i\sqrt{2})(2 + i\sqrt{2}) = 2^2 - (i\sqrt{2})^2 = 4 - (-2) = 4 + 2 = 6 \]

### Step 4: Calculate the numerator
The numerator becomes:
\[ 2 + i\sqrt{2} \]

### Step 5: Combine results
Thus, we have:
\[ \frac{2 + i\sqrt{2}}{6} = \frac{1}{3} + \frac{i\sqrt{2}}{6} \]

### Step 6: Identify the roots
Since \( \frac{1}{2 - i\sqrt{2}} \) is a root, the other root (by the property of complex conjugates) is:
\[ \frac{1}{2 + i\sqrt{2}} = \frac{2 - i\sqrt{2}}{6} = \frac{1}{3} - \frac{i\sqrt{2}}{6} \]

### Step 7: Sum and product of roots
Let the roots be \( r_1 = \frac{1}{3} + \frac{i\sqrt{2}}{6} \) and \( r_2 = \frac{1}{3} - \frac{i\sqrt{2}}{6} \).

The sum of the roots \( r_1 + r_2 \) is:
\[ \left(\frac{1}{3} + \frac{i\sqrt{2}}{6}\right) + \left(\frac{1}{3} - \frac{i\sqrt{2}}{6}\right) = \frac{2}{3} \]

The product of the roots \( r_1 \cdot r_2 \) is:
\[ \left(\frac{1}{3} + \frac{i\sqrt{2}}{6}\right)\left(\frac{1}{3} - \frac{i\sqrt{2}}{6}\right) = \left(\frac{1}{3}\right)^2 + \left(\frac{\sqrt{2}}{6}\right)^2 = \frac{1}{9} + \frac{2}{36} = \frac{1}{9} + \frac{1}{18} = \frac{2}{18} + \frac{1}{18} = \frac{3}{18} = \frac{1}{6} \]

### Step 8: Form the quadratic equation
Using the sum and product of the roots, we can form the quadratic equation:
\[ x^2 - \left(\frac{2}{3}\right)x + \frac{1}{6} = 0 \]

### Step 9: Clear the fractions
To eliminate the fractions, multiply through by 6:
\[ 6x^2 - 4x + 1 = 0 \]

### Step 10: Identify \( a \), \( b \), and \( c \)
From the equation \( 6x^2 - 4x + 1 = 0 \), we identify:
- \( a = 6 \)
- \( b = -4 \)
- \( c = 1 \)

### Final Answer
Thus, the values of \( a \), \( b \), and \( c \) are:
\[ a = 6, \quad b = -4, \quad c = 1 \]