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

Options

A. only real solution
B. no real solution
C. 4 real solutions
D. infinite number of real solutions

Answer

To solve the equation \( |2x - x^2 - 3| = 1 \), we will break it down into two cases based on the definition of absolute value.

### Step 1: Set up the two cases
The equation \( |A| = B \) can be rewritten as two separate equations:
1. \( A = B \)
2. \( A = -B \)

In our case, we have:
1. \( 2x - x^2 - 3 = 1 \)
2. \( 2x - x^2 - 3 = -1 \)

### Step 2: Solve the first case
For the first case:
\[ 2x - x^2 - 3 = 1 \]
Rearranging gives:
\[ -x^2 + 2x - 3 - 1 = 0 \implies -x^2 + 2x - 4 = 0 \implies x^2 - 2x + 4 = 0 \]

### Step 3: Calculate the discriminant for the first case
The discriminant \( D \) for the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
Here, \( a = 1, b = -2, c = 4 \):
\[ D = (-2)^2 - 4 \cdot 1 \cdot 4 = 4 - 16 = -12 \]
Since \( D < 0 \), this case has no real solutions.

### Step 4: Solve the second case
Now, for the second case:
\[ 2x - x^2 - 3 = -1 \]
Rearranging gives:
\[ -x^2 + 2x - 3 + 1 = 0 \implies -x^2 + 2x - 2 = 0 \implies x^2 - 2x + 2 = 0 \]

### Step 5: Calculate the discriminant for the second case
Again, we calculate the discriminant:
\[ D = b^2 - 4ac \]
Here, \( a = 1, b = -2, c = 2 \):
\[ D = (-2)^2 - 4 \cdot 1 \cdot 2 = 4 - 8 = -4 \]
Since \( D < 0 \), this case also has no real solutions.

### Conclusion
Since both cases yield no real solutions, we conclude that the equation \( |2x - x^2 - 3| = 1 \) has no real solutions.

### Final Answer
The equation has **no real solution**.

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