Solve the equation:
`5^(x+1)+5^(2-x)=5^(3)+1`

Answer

To solve the equation \( 5^{x+1} + 5^{2-x} = 5^3 + 1 \), we will follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ 5^{x+1} + 5^{2-x} = 5^3 + 1 \]
We know that \( 5^3 = 125 \), so we can rewrite the equation as:
\[ 5^{x+1} + 5^{2-x} = 126 \]

### Step 2: Use the properties of exponents
Using the property of exponents, we can rewrite \( 5^{x+1} \) and \( 5^{2-x} \):
\[ 5^{x+1} = 5^x \cdot 5^1 = 5 \cdot 5^x \]
\[ 5^{2-x} = \frac{5^2}{5^x} = \frac{25}{5^x} \]
Substituting these into the equation gives:
\[ 5 \cdot 5^x + \frac{25}{5^x} = 126 \]

### Step 3: Multiply through by \( 5^x \)
To eliminate the fraction, we can multiply the entire equation by \( 5^x \):
\[ 5 \cdot (5^x)^2 + 25 = 126 \cdot 5^x \]
This simplifies to:
\[ 5(5^x)^2 - 126(5^x) + 25 = 0 \]

### Step 4: Substitute \( t = 5^x \)
Let \( t = 5^x \). The equation now becomes:
\[ 5t^2 - 126t + 25 = 0 \]

### Step 5: Solve the quadratic equation
We can use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 5, b = -126, c = 25 \):
\[ b^2 - 4ac = (-126)^2 - 4 \cdot 5 \cdot 25 = 15876 - 500 = 15376 \]
Now, calculate \( \sqrt{15376} \):
\[ \sqrt{15376} = 124 \]
Now substituting back into the quadratic formula:
\[ t = \frac{126 \pm 124}{10} \]
Calculating the two possible values:
1. \( t = \frac{250}{10} = 25 \)
2. \( t = \frac{2}{10} = 0.2 \)

### Step 6: Back substitute to find \( x \)
Now we substitute back for \( t = 5^x \):
1. For \( t = 25 \):
\[ 5^x = 25 \implies 5^x = 5^2 \implies x = 2 \]
2. For \( t = 0.2 \):
\[ 5^x = 0.2 \implies 5^x = 5^{-1} \implies x = -1 \]

### Final Answer
Thus, the solutions for \( x \) are:
\[ x = -1 \quad \text{and} \quad x = 2 \]

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