The equation `|1-x| + x^(2) = 5` has :

Options

A. A rational root and an irrational root
B. Two rational roots
C. Two irrational roots
D. No real roots

Answer

To solve the equation \( |1 - x| + x^2 = 5 \), we need to consider two cases due to the absolute value.

### Step 1: Set up the cases based on the absolute value

The expression \( |1 - x| \) can be split into two cases:

1. **Case 1**: \( 1 - x \geq 0 \) (which implies \( x \leq 1 \))
- In this case, \( |1 - x| = 1 - x \).
- The equation becomes:
\[ 1 - x + x^2 = 5 \]

2. **Case 2**: \( 1 - x < 0 \) (which implies \( x > 1 \))
- In this case, \( |1 - x| = -(1 - x) = x - 1 \).
- The equation becomes:
\[ x - 1 + x^2 = 5 \]

### Step 2: Solve Case 1

For **Case 1**:
\[ 1 - x + x^2 = 5 \]
Rearranging gives:
\[ x^2 - x + 1 - 5 = 0 \implies x^2 - x - 4 = 0 \]
Now, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -1, c = -4 \):
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \]
\[ x = \frac{1 \pm \sqrt{1 + 16}}{2} = \frac{1 \pm \sqrt{17}}{2} \]

### Step 3: Solve Case 2

For **Case 2**:
\[ x - 1 + x^2 = 5 \]
Rearranging gives:
\[ x^2 + x - 1 - 5 = 0 \implies x^2 + x - 6 = 0 \]
Now, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = 1, c = -6 \):
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 + 24}}{2} = \frac{-1 \pm \sqrt{25}}{2} \]
\[ x = \frac{-1 \pm 5}{2} \]
Calculating the two roots:
1. \( x = \frac{4}{2} = 2 \)
2. \( x = \frac{-6}{2} = -3 \)

### Step 4: Collect the roots

From **Case 1**, the roots are:
\[ x = \frac{1 + \sqrt{17}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{17}}{2} \]
From **Case 2**, the roots are:
\[ x = 2 \quad \text{and} \quad x = -3 \]

### Step 5: Determine the nature of the roots

- The roots from Case 1 (\( \frac{1 + \sqrt{17}}{2} \) and \( \frac{1 - \sqrt{17}}{2} \)) are one rational and one irrational root.
- The roots from Case 2 (2 and -3) are both rational.

### Conclusion

Thus, the equation \( |1 - x| + x^2 = 5 \) has a total of **two rational roots and two irrational roots**.

### Final Answer:
The correct option is **(b) two rational roots**.

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