The probability that the roots of the equation `x^(2)+nx+(1)/(2)+(n)/(2)=0` are real where `n in N` such that `n le 5`, is

Options

A. `1//5`
B. `2//5`
C. `3//5`
D. `4//5`

Answer

To find the probability that the roots of the equation \( x^2 + nx + \left( \frac{1}{2} + \frac{n}{2} \right) = 0 \) are real, where \( n \in \mathbb{N} \) and \( n \leq 5 \), we can follow these steps:

### Step 1: Identify the condition for real roots
The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are real if the discriminant \( D \) is greater than or equal to zero. The discriminant is given by:
\[ D = b^2 - 4ac \]

### Step 2: Identify coefficients
For our equation:
- \( a = 1 \)
- \( b = n \)
- \( c = \frac{1}{2} + \frac{n}{2} = \frac{1 + n}{2} \)

### Step 3: Calculate the discriminant
Now, we can compute the discriminant:
\[ D = n^2 - 4 \cdot 1 \cdot \frac{1 + n}{2} \]
Simplifying this gives:
\[ D = n^2 - 2(1 + n) = n^2 - 2 - 2n \]
Rearranging this, we have:
\[ D = n^2 - 2n - 2 \]

### Step 4: Set the discriminant greater than or equal to zero
To find the values of \( n \) for which the roots are real, we need:
\[ n^2 - 2n - 2 \geq 0 \]

### Step 5: Solve the quadratic inequality
We can find the roots of the equation \( n^2 - 2n - 2 = 0 \) using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ = \frac{2 \pm \sqrt{4 + 8}}{2} = \frac{2 \pm \sqrt{12}}{2} = \frac{2 \pm 2\sqrt{3}}{2} = 1 \pm \sqrt{3} \]
Thus, the roots are:
\[ n_1 = 1 + \sqrt{3}, \quad n_2 = 1 - \sqrt{3} \]

### Step 6: Determine the intervals
The roots \( n_1 \) and \( n_2 \) divide the number line into intervals. We need to check the sign of \( D \) in these intervals:
- For \( n < 1 - \sqrt{3} \)
- For \( 1 - \sqrt{3} < n < 1 + \sqrt{3} \)
- For \( n > 1 + \sqrt{3} \)

Since \( 1 - \sqrt{3} \) is negative, we only consider \( n \geq 1 \).

### Step 7: Evaluate for \( n = 1, 2, 3, 4, 5 \)
Now we check the values of \( n \) from 1 to 5:

1. **For \( n = 1 \)**:
\[ D = 1^2 - 2(1) - 2 = 1 - 2 - 2 = -3 \quad (\text{Not real}) \]

2. **For \( n = 2 \)**:
\[ D = 2^2 - 2(2) - 2 = 4 - 4 - 2 = -2 \quad (\text{Not real}) \]

3. **For \( n = 3 \)**:
\[ D = 3^2 - 2(3) - 2 = 9 - 6 - 2 = 1 \quad (\text{Real}) \]

4. **For \( n = 4 \)**:
\[ D = 4^2 - 2(4) - 2 = 16 - 8 - 2 = 6 \quad (\text{Real}) \]

5. **For \( n = 5 \)**:
\[ D = 5^2 - 2(5) - 2 = 25 - 10 - 2 = 13 \quad (\text{Real}) \]

### Step 8: Count favorable outcomes
The values of \( n \) that yield real roots are \( n = 3, 4, 5 \). Thus, there are 3 favorable outcomes.

### Step 9: Calculate the probability
The total possible outcomes for \( n \) (natural numbers from 1 to 5) is 5. Therefore, the probability \( P \) that the roots are real is:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{3}{5} \]

### Final Answer
The probability that the roots of the equation are real is \( \frac{3}{5} \).