The equation `(x^(2))/(1-|x-2|)=1` has

Options

A. one real solution
B. two real solutions
C. three real solutions
D. no real solution

Answer

To solve the equation \(\frac{x^2}{1 - |x - 2|} = 1\), we need to consider two cases based on the definition of the absolute value function.

### Step 1: Identify Cases for Absolute Value
The expression \(|x - 2|\) can be defined in two cases:
1. **Case 1:** \(x - 2 \geq 0\) (i.e., \(x \geq 2\))
2. **Case 2:** \(x - 2 < 0\) (i.e., \(x < 2\))

### Step 2: Solve Case 1 (\(x \geq 2\))
In this case, \(|x - 2| = x - 2\). Therefore, the equation becomes:
\[ \frac{x^2}{1 - (x - 2)} = 1 \]
This simplifies to:
\[ \frac{x^2}{3 - x} = 1 \]
Multiplying both sides by \(3 - x\) (assuming \(3 - x \neq 0\)):
\[ x^2 = 3 - x \]
Rearranging gives:
\[ x^2 + x - 3 = 0 \]

### Step 3: Solve the Quadratic Equation
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = 1\), and \(c = -3\):
\[ D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-3) = 1 + 12 = 13 \]
Thus, the roots are:
\[ x = \frac{-1 \pm \sqrt{13}}{2} \]
Calculating the roots:
1. \(x_1 = \frac{-1 + \sqrt{13}}{2}\)
2. \(x_2 = \frac{-1 - \sqrt{13}}{2}\)

### Step 4: Check Validity of Roots for Case 1
We need to check if these roots satisfy \(x \geq 2\):
- For \(x_1\):
\[ x_1 = \frac{-1 + \sqrt{13}}{2} \approx 1.30 \quad (\text{not valid}) \]
- For \(x_2\):
\[ x_2 = \frac{-1 - \sqrt{13}}{2} \approx -2.30 \quad (\text{not valid}) \]

### Step 5: Solve Case 2 (\(x < 2\))
In this case, \(|x - 2| = -(x - 2) = 2 - x\). Therefore, the equation becomes:
\[ \frac{x^2}{1 - (2 - x)} = 1 \]
This simplifies to:
\[ \frac{x^2}{x - 1} = 1 \]
Multiplying both sides by \(x - 1\) (assuming \(x - 1 \neq 0\)):
\[ x^2 = x - 1 \]
Rearranging gives:
\[ x^2 - x + 1 = 0 \]

### Step 6: Solve the Quadratic Equation for Case 2
Using the quadratic formula:
\[ D = (-1)^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]
Since the discriminant \(D < 0\), there are no real solutions for this case.

### Conclusion
Combining the results from both cases, we find that the equation \(\frac{x^2}{1 - |x - 2|} = 1\) has no real solutions.