Suppose a is an integer and `x_(1) and x_(2)` are positive real roots of `x^(2) + (2a - 1) x + a^(2) = 0`, then value of `|sqrt(x_(1))-sqrt(x_(2))|` is

Options

A. 1
B. 2
C. a
D. 1-4a

Answer

To solve the problem, we need to find the value of \(|\sqrt{x_1} - \sqrt{x_2}|\) given that \(x_1\) and \(x_2\) are the positive real roots of the quadratic equation \(x^2 + (2a - 1)x + a^2 = 0\).

### Step-by-Step Solution:

1. **Identify the coefficients of the quadratic equation:**
The given quadratic equation is:
\[ x^2 + (2a - 1)x + a^2 = 0 \]
Here, \(A = 1\), \(B = 2a - 1\), and \(C = a^2\).

2. **Use the quadratic formula to find the roots:**
The roots of a quadratic equation \(Ax^2 + Bx + C = 0\) can be found using the formula:
\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Substituting the values of \(A\), \(B\), and \(C\):
\[ x_1, x_2 = \frac{-(2a - 1) \pm \sqrt{(2a - 1)^2 - 4 \cdot 1 \cdot a^2}}{2 \cdot 1} \]

3. **Calculate the discriminant:**
The discriminant \(D\) is given by:
\[ D = (2a - 1)^2 - 4a^2 \]
Simplifying this:
\[ D = 4a^2 - 4a + 1 - 4a^2 = -4a + 1 \]

4. **Roots of the quadratic equation:**
The roots can be expressed as:
\[ x_1, x_2 = \frac{1 - 2a \pm \sqrt{1 - 4a}}{2} \]

5. **Calculate the values of \(\sqrt{x_1}\) and \(\sqrt{x_2}\):**
We need to find \(|\sqrt{x_1} - \sqrt{x_2}|\). Using the identity:
\[ |\sqrt{x_1} - \sqrt{x_2}| = \frac{|x_1 - x_2|}{|\sqrt{x_1} + \sqrt{x_2}|} \]

6. **Find \(x_1 + x_2\) and \(x_1 - x_2\):**
From Vieta's formulas:
- \(x_1 + x_2 = -\frac{B}{A} = 1 - 2a\)
- \(x_1 x_2 = \frac{C}{A} = a^2\)

Now, to find \(x_1 - x_2\):
\[ x_1 - x_2 = \sqrt{D} = \sqrt{-4a + 1} \]

7. **Calculate \(|\sqrt{x_1} - \sqrt{x_2}|\):**
Substituting into the earlier expression:
\[ |\sqrt{x_1} - \sqrt{x_2}| = \frac{\sqrt{-4a + 1}}{|\sqrt{x_1} + \sqrt{x_2}|} \]
We know:
\[ \sqrt{x_1} + \sqrt{x_2} = \sqrt{(x_1 + x_2) + 2\sqrt{x_1 x_2}} = \sqrt{(1 - 2a) + 2a} = \sqrt{1} \]
Thus:
\[ |\sqrt{x_1} + \sqrt{x_2}| = 1 \]

8. **Final Calculation:**
Therefore:
\[ |\sqrt{x_1} - \sqrt{x_2}| = \sqrt{-4a + 1} \]

9. **Conclusion:**
Since \(a\) is an integer, we find that the expression simplifies to:
\[ |\sqrt{x_1} - \sqrt{x_2}| = 1 \]

### Final Answer:
\[ |\sqrt{x_1} - \sqrt{x_2}| = 1 \]