Find the number of solutions of the equation `(x^(2))/(1-|x-2|)=1`, graphically.

Answer

To find the number of solutions of the equation \(\frac{x^2}{1 - |x - 2|} = 1\) graphically, we will follow these steps:

### Step 1: Rewrite the equation
First, we can rewrite the equation by cross-multiplying:
\[ x^2 = 1 - |x - 2| \]
This gives us two cases to consider based on the definition of the absolute value.

### Step 2: Case 1 - \(x - 2 \geq 0\) (i.e., \(x \geq 2\))
In this case, \(|x - 2| = x - 2\). Substituting this into the equation gives:
\[ x^2 = 1 - (x - 2) \]
Simplifying this, we have:
\[ x^2 = 3 - x \]
Rearranging gives us:
\[ x^2 + x - 3 = 0 \]

### Step 3: Find the discriminant for Case 1
To determine the number of solutions, we calculate the discriminant \(D\):
\[ D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-3) = 1 + 12 = 13 \]
Since \(D > 0\), there are two real solutions for this case.

### Step 4: Case 2 - \(x - 2 < 0\) (i.e., \(x < 2\))
In this case, \(|x - 2| = -(x - 2) = 2 - x\). Substituting this into the equation gives:
\[ x^2 = 1 - (2 - x) \]
Simplifying this, we have:
\[ x^2 = x - 1 \]
Rearranging gives us:
\[ x^2 - x + 1 = 0 \]

### Step 5: Find the discriminant for Case 2
Now, we calculate the discriminant \(D\):
\[ D = (-1)^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]
Since \(D < 0\), there are no real solutions for this case.

### Step 6: Combine the results
From Case 1, we found 2 solutions, and from Case 2, we found 0 solutions. Therefore, the total number of solutions to the original equation is:
\[ \text{Total solutions} = 2 + 0 = 2 \]

### Final Answer
The number of solutions of the equation \(\frac{x^2}{1 - |x - 2|} = 1\) is **2**.

---