Solution of the equation `3^(2x^2)-2.3^(x^2+x+6)+3^(2(x+6))=0` is

Options

A. a) `{-3,2}`
B. b) `{6,-1}`
C. c) `{-2,3}`
D. d) `{1,-6}`

Explanation

To solve the equation \( 3^{2x^2} - 2 \cdot 3^{x^2 + x + 6} + 3^{2(x + 6)} = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start by rewriting the equation in a more manageable form. We can express \( 3^{2x^2} \) and \( 3^{2(x + 6)} \) as squares:
\[ 3^{2x^2} - 2 \cdot 3^{x^2 + x + 6} + 3^{2(x + 6)} = 0 \]
This can be rewritten as:
\[ (3^{x^2})^2 - 2 \cdot 3^{x^2} \cdot 3^{x + 6} + (3^{x + 6})^2 = 0 \]

### Step 2: Apply the quadratic formula
Let \( a = 3^{x^2} \) and \( b = 3^{x + 6} \). The equation now resembles the form \( a^2 - 2ab + b^2 = 0 \), which can be factored as:
\[ (a - b)^2 = 0 \]
Thus, we have:
\[ 3^{x^2} - 3^{x + 6} = 0 \]

### Step 3: Set the exponents equal
From the equation \( 3^{x^2} = 3^{x + 6} \), we can equate the exponents:
\[ x^2 = x + 6 \]

### Step 4: Rearrange into standard form
Rearranging gives us:
\[ x^2 - x - 6 = 0 \]

### Step 5: Factor the quadratic
Next, we can factor the quadratic equation:
\[ (x - 3)(x + 2) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us the solutions:
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

### Final Answer
Thus, the solutions to the equation are:
\[ x = 3 \quad \text{and} \quad x = -2 \]

### Conclusion
The correct option is \( -2, 3 \).

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