The root of the equation `3^(2x)-10.3^(x)+9=0` are

Options

A. 1,2
B. 0,2
C. 0,1
D. 1,3

Answer

To solve the equation \(3^{2x} - 10 \cdot 3^x + 9 = 0\), we can follow these steps:

### Step 1: Substitute \(t\) for \(3^x\)
Let \(t = 3^x\). Then, we can rewrite \(3^{2x}\) as \(t^2\). The equation now becomes:
\[ t^2 - 10t + 9 = 0 \]

### Step 2: Factor the quadratic equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to \(9\) (the constant term) and add up to \(-10\) (the coefficient of \(t\)). The numbers \(-1\) and \(-9\) satisfy these conditions. Thus, we can factor the equation as:
\[ (t - 1)(t - 9) = 0 \]

### Step 3: Solve for \(t\)
Setting each factor to zero gives us:
\[ t - 1 = 0 \quad \Rightarrow \quad t = 1 \]
\[ t - 9 = 0 \quad \Rightarrow \quad t = 9 \]

### Step 4: Substitute back to find \(x\)
Now, we substitute back \(t = 3^x\):
1. For \(t = 1\):
\[ 3^x = 1 \quad \Rightarrow \quad x = 0 \quad (\text{since } 3^0 = 1) \]
2. For \(t = 9\):
\[ 3^x = 9 \quad \Rightarrow \quad x = 2 \quad (\text{since } 3^2 = 9) \]

### Conclusion
The roots of the equation \(3^{2x} - 10 \cdot 3^x + 9 = 0\) are:
\[ x = 0 \quad \text{and} \quad x = 2 \]

### Final Answer
Thus, the roots are \(0\) and \(2\).

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