Solve `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
Start with the given equation:
\[ 5^{(x+1)} + 5^{(2-x)} = 5^3 + 1 \]

### Step 2: Simplify the right side
Calculate \( 5^3 + 1 \):
\[ 5^3 = 125 \quad \text{so,} \quad 5^3 + 1 = 125 + 1 = 126 \]
Now, the equation becomes:
\[ 5^{(x+1)} + 5^{(2-x)} = 126 \]

### Step 3: Substitute \( 5^x \) with \( y \)
Let \( 5^x = y \). Then \( 5^{(x+1)} = 5 \cdot y \) and \( 5^{(2-x)} = \frac{25}{y} \) (since \( 5^{(2-x)} = 5^2 \cdot 5^{-x} = \frac{25}{y} \)).
Substituting these into the equation gives:
\[ 5y + \frac{25}{y} = 126 \]

### Step 4: Multiply through by \( y \)
To eliminate the fraction, multiply the entire equation by \( y \):
\[ 5y^2 + 25 = 126y \]

### Step 5: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ 5y^2 - 126y + 25 = 0 \]

### Step 6: Apply the quadratic formula
The quadratic formula is given by:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation, \( a = 5 \), \( b = -126 \), and \( c = 25 \). Plugging in these values:
\[ y = \frac{126 \pm \sqrt{(-126)^2 - 4 \cdot 5 \cdot 25}}{2 \cdot 5} \]
Calculating the discriminant:
\[ (-126)^2 = 15876 \quad \text{and} \quad 4 \cdot 5 \cdot 25 = 500 \]
Thus:
\[ 15876 - 500 = 15376 \]
Now, taking the square root:
\[ \sqrt{15376} = 124 \]
So, substituting back into the formula gives:
\[ y = \frac{126 \pm 124}{10} \]

### Step 7: Calculate the two possible values for \( y \)
Calculating the two cases:
1. \( y = \frac{126 + 124}{10} = \frac{250}{10} = 25 \)
2. \( y = \frac{126 - 124}{10} = \frac{2}{10} = \frac{1}{5} \)

### Step 8: Solve for \( x \)
Recall that \( y = 5^x \):
1. For \( y = 25 \):
\[ 5^x = 25 \implies 5^x = 5^2 \implies x = 2 \]
2. For \( y = \frac{1}{5} \):
\[ 5^x = \frac{1}{5} \implies 5^x = 5^{-1} \implies x = -1 \]

### Final Solution
The solutions for \( x \) are:
\[ \boxed{2 \text{ and } -1} \]