If `a^2 + b^2 + c^2 + 1/a^2 + 1/b^2 + 1/c^2 = 6`, then what is the value of `a^2 + b^2 + c^2` ?

Options

A. 3
B. 6
C. `-3`
D. 2

Answer

To solve the equation \( a^2 + b^2 + c^2 + \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} = 6 \) and find the value of \( a^2 + b^2 + c^2 \), we can follow these steps:

### Step 1: Assume \( a = b = c \)
To simplify the problem, let's assume \( a = b = c = x \). This means we can rewrite the equation in terms of \( x \):
\[ 3x^2 + 3 \cdot \frac{1}{x^2} = 6 \]

### Step 2: Simplify the equation
Now, we can factor out the 3 from the left side:
\[ 3 \left( x^2 + \frac{1}{x^2} \right) = 6 \]
Dividing both sides by 3 gives:
\[ x^2 + \frac{1}{x^2} = 2 \]

### Step 3: Solve for \( x^2 \)
To solve for \( x^2 \), we can multiply both sides by \( x^2 \):
\[ x^4 - 2x^2 + 1 = 0 \]
Let \( y = x^2 \). The equation becomes:
\[ y^2 - 2y + 1 = 0 \]
Factoring gives:
\[ (y - 1)^2 = 0 \]
Thus, \( y - 1 = 0 \) implies:
\[ y = 1 \]
So, \( x^2 = 1 \).

### Step 4: Find \( a^2 + b^2 + c^2 \)
Since \( a^2 = b^2 = c^2 = x^2 = 1 \), we can now find:
\[ a^2 + b^2 + c^2 = 1 + 1 + 1 = 3 \]

### Final Answer
Thus, the value of \( a^2 + b^2 + c^2 \) is:
\[ \boxed{3} \]