The values of x satisfying the equation
`5^(2x)-5^(x+3)+125=5^(x)` are

Options

A. 0 and 2
B. `-1` and 3
C. 0 and -3
D. 0 and 3

Answer

To solve the equation \( 5^{2x} - 5^{x+3} + 125 = 5^{x} \), we can follow these steps:

### Step 1: Substitute \( y = 5^x \)
Let \( y = 5^x \). Then, we can rewrite \( 5^{2x} \) as \( (5^x)^2 = y^2 \) and \( 5^{x+3} \) as \( 5^x \cdot 5^3 = y \cdot 125 \).

### Step 2: Rewrite the equation
Substituting these values into the equation gives us:
\[ y^2 - 125y + 125 = y \]

### Step 3: Rearrange the equation
Now, rearranging the equation to set it to zero:
\[ y^2 - 125y + 125 - y = 0 \]
This simplifies to:
\[ y^2 - 126y + 125 = 0 \]

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation \( y^2 - 126y + 125 = 0 \). We are looking for two numbers that multiply to \( 125 \) and add up to \( -126 \). The numbers are \( -125 \) and \( -1 \).

### Step 5: Write the factored form
Thus, we can write the equation as:
\[ (y - 125)(y - 1) = 0 \]

### Step 6: Solve for \( y \)
Setting each factor to zero gives us:
1. \( y - 125 = 0 \) → \( y = 125 \)
2. \( y - 1 = 0 \) → \( y = 1 \)

### Step 7: Substitute back for \( x \)
Recall that \( y = 5^x \). Therefore, we have:
1. \( 5^x = 125 \)
2. \( 5^x = 1 \)

### Step 8: Solve for \( x \)
For \( 5^x = 125 \):
\[ 125 = 5^3 \implies x = 3 \]

For \( 5^x = 1 \):
\[ 1 = 5^0 \implies x = 0 \]

### Conclusion
The values of \( x \) satisfying the equation are:
\[ x = 0 \quad \text{and} \quad x = 3 \]

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