The number of solutions of the equation `125^(x)+45^(x)=2.27^(x)` is

Options

A. `1`
B. `2`
C. `0`
D. more than two

Answer

To solve the equation \( 125^x + 45^x = 2 \cdot 27^x \), we will follow these steps:

### Step 1: Rewrite the bases in terms of prime factors
We can express the bases as follows:
- \( 125 = 5^3 \)
- \( 45 = 9 \cdot 5 = 3^2 \cdot 5 \)
- \( 27 = 3^3 \)

So, we can rewrite the equation:
\[ (5^3)^x + (3^2 \cdot 5)^x = 2 \cdot (3^3)^x \]
This simplifies to:
\[ 5^{3x} + 3^{2x} \cdot 5^x = 2 \cdot 3^{3x} \]

### Step 2: Divide the entire equation by \( 3^{3x} \)
To simplify the equation further, we divide each term by \( 3^{3x} \):
\[ \frac{5^{3x}}{3^{3x}} + \frac{3^{2x} \cdot 5^x}{3^{3x}} = 2 \]
This gives us:
\[ \left(\frac{5}{3}\right)^{3x} + \left(\frac{5}{3}\right)^x \cdot \left(\frac{3}{3}\right)^{2x} = 2 \]
Which simplifies to:
\[ \left(\frac{5}{3}\right)^{3x} + \left(\frac{5}{3}\right)^x = 2 \]

### Step 3: Let \( t = \left(\frac{5}{3}\right)^x \)
Now, we can substitute \( t \) for \( \left(\frac{5}{3}\right)^x \):
\[ t^3 + t = 2 \]
Rearranging gives us:
\[ t^3 + t - 2 = 0 \]

### Step 4: Find the roots of the cubic equation
We can try to find rational roots using the Rational Root Theorem. Testing \( t = 1 \):
\[ 1^3 + 1 - 2 = 0 \]
So, \( t = 1 \) is a root. We can factor the cubic polynomial:
\[ t^3 + t - 2 = (t - 1)(t^2 + t + 2) \]

### Step 5: Solve the quadratic equation
Now we need to solve the quadratic equation \( t^2 + t + 2 = 0 \) using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 - 8}}{2} = \frac{-1 \pm \sqrt{-7}}{2} \]
This gives us complex roots, which means there are no real solutions from this quadratic.

### Step 6: Find the number of solutions
Since the only real solution we found is \( t = 1 \), we substitute back to find \( x \):
\[ \left(\frac{5}{3}\right)^x = 1 \implies x = 0 \]
Thus, the equation has exactly **one solution**.

### Final Answer
The number of solutions of the equation \( 125^x + 45^x = 2 \cdot 27^x \) is **1**.

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