The equation `|x-x^(2)-1|=|2x-3-x^(2)|` has

Options

A. one solution
B. two solutions
C. no solution
D. infinite solutions

Answer

To solve the equation \( |x - x^2 - 1| = |2x - 3 - x^2| \), we will consider the cases for the absolute values.

### Step 1: Set up the cases based on the absolute values.

The equation can be split into four cases based on the definitions of the absolute values:

1. \( x - x^2 - 1 = 2x - 3 - x^2 \)
2. \( x - x^2 - 1 = -(2x - 3 - x^2) \)
3. \( -(x - x^2 - 1) = 2x - 3 - x^2 \)
4. \( -(x - x^2 - 1) = -(2x - 3 - x^2) \)

### Step 2: Solve the first case.

**Case 1:** \( x - x^2 - 1 = 2x - 3 - x^2 \)

Rearranging gives:
\[ x - x^2 - 1 = 2x - 3 - x^2 \]
Cancelling \( -x^2 \) from both sides:
\[ x - 1 = 2x - 3 \]
Rearranging gives:
\[ -1 + 3 = 2x - x \]
\[ 2 = x \]

### Step 3: Solve the second case.

**Case 2:** \( x - x^2 - 1 = -(2x - 3 - x^2) \)

Rearranging gives:
\[ x - x^2 - 1 = -2x + 3 + x^2 \]
Combining like terms:
\[ x - x^2 + 2x - x^2 = 3 + 1 \]
\[ 3x - 2x^2 = 4 \]
Rearranging gives:
\[ 2x^2 - 3x + 4 = 0 \]

Now, we will find the discriminant:
\[ D = b^2 - 4ac = (-3)^2 - 4 \cdot 2 \cdot 4 = 9 - 32 = -23 \]
Since the discriminant is negative, there are no real solutions from this case.

### Step 4: Solve the third case.

**Case 3:** \( -(x - x^2 - 1) = 2x - 3 - x^2 \)

This simplifies to:
\[ -x + x^2 + 1 = 2x - 3 - x^2 \]
Combining like terms:
\[ x^2 + x + 1 + 3 = 2x \]
Rearranging gives:
\[ x^2 - x + 4 = 0 \]
Finding the discriminant:
\[ D = (-1)^2 - 4 \cdot 1 \cdot 4 = 1 - 16 = -15 \]
Again, the discriminant is negative, so there are no real solutions from this case.

### Step 5: Solve the fourth case.

**Case 4:** \( -(x - x^2 - 1) = -(2x - 3 - x^2) \)

This simplifies to:
\[ -x + x^2 + 1 = -2x + 3 + x^2 \]
Cancelling \( x^2 \) gives:
\[ -x + 1 = -2x + 3 \]
Rearranging gives:
\[ -x + 2x = 3 - 1 \]
\[ x = 2 \]

### Conclusion

From the analysis of all four cases, we find that the only solution is \( x = 2 \). Thus, the equation \( |x - x^2 - 1| = |2x - 3 - x^2| \) has **one solution**.