Solve for `x:3^(x+2)+3^(-x)=10`

Answer

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

### Step 1: Rewrite the equation
Start by rewriting the equation:
\[ 3^{(x+2)} + 3^{-x} = 10 \]
We can express \( 3^{(x+2)} \) as \( 3^x \cdot 3^2 \):
\[ 3^x \cdot 9 + 3^{-x} = 10 \]

### Step 2: Substitute \( 3^x \)
Let \( a = 3^x \). Then, we can rewrite the equation:
\[ 9a + \frac{1}{a} = 10 \]

### Step 3: Multiply through by \( a \)
To eliminate the fraction, multiply the entire equation by \( a \):
\[ 9a^2 + 1 = 10a \]

### Step 4: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ 9a^2 - 10a + 1 = 0 \]

### Step 5: Use the quadratic formula
We can solve for \( a \) using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 9 \), \( b = -10 \), and \( c = 1 \):
\[ a = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 9 \cdot 1}}{2 \cdot 9} \]
\[ = \frac{10 \pm \sqrt{100 - 36}}{18} \]
\[ = \frac{10 \pm \sqrt{64}}{18} \]
\[ = \frac{10 \pm 8}{18} \]

### Step 6: Calculate the values of \( a \)
This gives us two possible values for \( a \):
1. \( a = \frac{18}{18} = 1 \)
2. \( a = \frac{2}{18} = \frac{1}{9} \)

### Step 7: Substitute back for \( x \)
Recall that \( a = 3^x \). We can now substitute back to find \( x \):
1. For \( a = 1 \):
\[ 3^x = 1 \implies x = 0 \quad (\text{since } 3^0 = 1) \]
2. For \( a = \frac{1}{9} \):
\[ 3^x = \frac{1}{9} \implies 3^x = 3^{-2} \implies x = -2 \]

### Final Solution
Thus, the solutions for \( x \) are:
\[ x = 0 \quad \text{and} \quad x = -2 \]

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