The number of real solutions of the equation `((5)/(7))^(2)=-x^(2)+2x-3` is equal to

Options

A. 0
B. 1
C. 2
D. none of these

Answer

To solve the equation \(\left(\frac{5}{7}\right)^2 = -x^2 + 2x - 3\) and determine the number of real solutions, we can follow these steps:

### Step 1: Simplify the Left-Hand Side
Calculate \(\left(\frac{5}{7}\right)^2\):
\[ \left(\frac{5}{7}\right)^2 = \frac{25}{49} \]

### Step 2: Rewrite the Equation
Now we can rewrite the equation:
\[ \frac{25}{49} = -x^2 + 2x - 3 \]

### Step 3: Move All Terms to One Side
To form a standard quadratic equation, move all terms to one side:
\[ 0 = -x^2 + 2x - 3 - \frac{25}{49} \]
To combine the constants, convert \(-3\) into a fraction:
\[ -3 = -\frac{147}{49} \]
Now, the equation becomes:
\[ 0 = -x^2 + 2x - \frac{147}{49} - \frac{25}{49} \]
Combine the constants:
\[ 0 = -x^2 + 2x - \frac{172}{49} \]

### Step 4: Rearrange the Equation
Multiply through by \(-1\) to make the \(x^2\) term positive:
\[ x^2 - 2x + \frac{172}{49} = 0 \]

### Step 5: Identify Coefficients
In the standard form \(ax^2 + bx + c = 0\), we have:
- \(a = 1\)
- \(b = -2\)
- \(c = \frac{172}{49}\)

### Step 6: Calculate the Discriminant
The discriminant \(D\) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values:
\[ D = (-2)^2 - 4 \cdot 1 \cdot \frac{172}{49} \]
Calculating \(D\):
\[ D = 4 - \frac{688}{49} \]
Convert \(4\) into a fraction:
\[ D = \frac{196}{49} - \frac{688}{49} = \frac{196 - 688}{49} = \frac{-492}{49} \]

### Step 7: Analyze the Discriminant
Since \(D < 0\), this indicates that there are no real solutions to the equation.

### Conclusion
The number of real solutions of the equation \(\left(\frac{5}{7}\right)^2 = -x^2 + 2x - 3\) is:
\[ \text{0} \]