The solution set of `((3)/(5))^(x)=x-x^(2)-9` is

Options

A. {0}
B. {1}
C. `phi`
D. none of these

Answer

To solve the equation \(\left(\frac{3}{5}\right)^{x} = x - x^{2} - 9\), we will analyze both sides of the equation by graphing and determining the points of intersection.

### Step 1: Rewrite the equation
We start with the equation:
\[ \left(\frac{3}{5}\right)^{x} = x - x^{2} - 9 \]

### Step 2: Analyze the left-hand side
The left-hand side, \(\left(\frac{3}{5}\right)^{x}\), is an exponential function. Since \(\frac{3}{5} < 1\), this function will be decreasing. As \(x\) approaches infinity, \(\left(\frac{3}{5}\right)^{x}\) approaches \(0\), and as \(x\) approaches negative infinity, \(\left(\frac{3}{5}\right)^{x}\) approaches infinity.

### Step 3: Analyze the right-hand side
The right-hand side, \(x - x^{2} - 9\), is a quadratic function. We can rewrite it as:
\[ -x^{2} + x - 9 \]
This is a downward-opening parabola (since the coefficient of \(x^{2}\) is negative).

### Step 4: Find the vertex of the quadratic
The vertex of a quadratic \(ax^2 + bx + c\) is given by the formula \(x = -\frac{b}{2a}\). Here, \(a = -1\) and \(b = 1\):
\[ x = -\frac{1}{2 \cdot -1} = \frac{1}{2} \]
Now we can find the value of the quadratic at \(x = \frac{1}{2}\):
\[ y = -\left(\frac{1}{2}\right)^{2} + \frac{1}{2} - 9 = -\frac{1}{4} + \frac{1}{2} - 9 = -\frac{1}{4} + \frac{2}{4} - \frac{36}{4} = -\frac{35}{4} \]

### Step 5: Determine the y-intercept
To find the y-intercept of the quadratic, we set \(x = 0\):
\[ y = 0 - 0 - 9 = -9 \]

### Step 6: Analyze the discriminant
To find the number of real roots of the quadratic \(x - x^{2} - 9\), we calculate the discriminant \(D = b^{2} - 4ac\):
\[ D = 1^{2} - 4(-1)(-9) = 1 - 36 = -35 \]
Since \(D < 0\), the quadratic does not intersect the x-axis, meaning it has no real roots.

### Step 7: Conclusion
Since the left-hand side \(\left(\frac{3}{5}\right)^{x}\) approaches \(0\) as \(x\) increases and the right-hand side \(x - x^{2} - 9\) is always less than \(0\) (as shown by the vertex and y-intercept), there are no points of intersection between the two graphs. Therefore, the solution set is empty.

The final answer is:
\[ \text{Solution set: } \emptyset \]