Sum of all values of x satisfying the equation `25^(2x-x^2+1)+9^(2x-x^2+1)=34(15^(2x-x^2))` is:

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equation \( 25^{(2x - x^2 + 1)} + 9^{(2x - x^2 + 1)} = 34 \cdot 15^{(2x - x^2)} \), we can follow these steps:

### Step 1: Rewrite the equation in terms of powers
We can express \( 25 \) and \( 9 \) as powers of \( 5 \) and \( 3 \) respectively:
\[ 25 = 5^2 \quad \text{and} \quad 9 = 3^2 \]
Thus, we can rewrite the equation as:
\[ (5^2)^{(2x - x^2 + 1)} + (3^2)^{(2x - x^2 + 1)} = 34 \cdot (15^{(2x - x^2)}) \]
This simplifies to:
\[ 5^{(4x - 2x^2 + 2)} + 3^{(4x - 2x^2 + 2)} = 34 \cdot 15^{(2x - x^2)} \]

### Step 2: Substitute \( y = 2x - x^2 \)
Let \( y = 2x - x^2 \). Then we can rewrite the equation as:
\[ 5^{(y + 1)} + 3^{(y + 1)} = 34 \cdot 15^y \]
This can be simplified to:
\[ 5 \cdot 5^y + 3 \cdot 3^y = 34 \cdot 15^y \]

### Step 3: Analyze the equation
Notice that \( 15 = 5 \cdot 3 \), so we can rewrite \( 15^y \) as:
\[ 15^y = 5^y \cdot 3^y \]
Thus, we have:
\[ 5^{y + 1} + 3^{y + 1} = 34 \cdot (5^y \cdot 3^y) \]

### Step 4: Set \( 5^y = a \) and \( 3^y = b \)
Let \( a = 5^y \) and \( b = 3^y \). The equation becomes:
\[ 5a + 3b = 34ab \]

### Step 5: Rearranging the equation
Rearranging gives us:
\[ 34ab - 5a - 3b = 0 \]
This can be factored as:
\[ (34b - 5)a - 3b = 0 \]

### Step 6: Solve for \( a \) and \( b \)
This gives us two cases:
1. \( a = 0 \) (which is not possible since \( a = 5^y \))
2. \( 34b - 5 = 0 \) which leads to \( b = \frac{5}{34} \)

### Step 7: Substitute back to find \( y \)
From \( b = 3^y \):
\[ 3^y = \frac{5}{34} \]
Taking logarithm:
\[ y \log(3) = \log\left(\frac{5}{34}\right) \]
Thus,
\[ y = \frac{\log\left(\frac{5}{34}\right)}{\log(3)} \]

### Step 8: Substitute \( y \) back to find \( x \)
Recall \( y = 2x - x^2 \):
\[ x^2 - 2x + y = 0 \]
Using the quadratic formula, we find:
\[ x = \frac{2 \pm \sqrt{4 - 4y}}{2} \]
This simplifies to:
\[ x = 1 \pm \sqrt{1 - y} \]

### Step 9: Calculate the sum of all values of \( x \)
The roots of the quadratic equation sum to \( 2 \) (since \( x_1 + x_2 = 2 \) from the quadratic formula).

### Final Answer
Thus, the sum of all values of \( x \) satisfying the equation is:
\[ \boxed{2} \]