For real roots, the solution of the equation `2^(x^(2)): 2^(2x)=8:1` is

Options

A. 1,2
B. 2,3
C. 3,-1
D. none

Answer

To solve the equation \( \frac{2^{x^2}}{2^{2x}} = \frac{8}{1} \) for real roots, we can follow these steps:

### Step 1: Simplify the equation
We start with the equation:

\[ \frac{2^{x^2}}{2^{2x}} = 8 \]

Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify the left side:

\[ 2^{x^2 - 2x} = 8 \]

### Step 2: Rewrite 8 as a power of 2
Next, we rewrite 8 as a power of 2:

\[ 8 = 2^3 \]

So, we can rewrite the equation as:

\[ 2^{x^2 - 2x} = 2^3 \]

### Step 3: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:

\[ x^2 - 2x = 3 \]

### Step 4: Rearrange the equation
Rearranging the equation gives us a standard quadratic form:

\[ x^2 - 2x - 3 = 0 \]

### Step 5: Factor the quadratic equation
Now we will factor the quadratic equation:

\[ x^2 - 2x - 3 = (x - 3)(x + 1) = 0 \]

### Step 6: Solve for x
Setting each factor equal to zero gives us the solutions:

1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x + 1 = 0 \) → \( x = -1 \)

### Step 7: Conclusion
Thus, the solutions for the equation are:

\[ x = 3 \quad \text{and} \quad x = -1 \]