If the roots of the quadratic equation `x^(2) +2x+k =0` are real, then

Options

A. `k lt 0`
B. `k le 0`
C. `k lt 1`
D. `k le 1`

Answer

To determine the condition for the roots of the quadratic equation \(x^2 + 2x + k = 0\) to be real, we need to analyze the discriminant of the equation. The discriminant (\(D\)) is given by the formula:

\[ D = b^2 - 4ac \]

where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).

### Step-by-step Solution:

1. **Identify the coefficients:**
- From the equation \(x^2 + 2x + k = 0\), we have:
- \(a = 1\)
- \(b = 2\)
- \(c = k\)

2. **Write the discriminant:**
- Substitute the coefficients into the discriminant formula:
\[ D = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot k \]

3. **Simplify the discriminant:**
- Calculate \(b^2\):
\[ D = 4 - 4k \]

4. **Set the discriminant condition for real roots:**
- For the roots to be real, the discriminant must be greater than or equal to zero:
\[ 4 - 4k \geq 0 \]

5. **Solve the inequality:**
- Rearranging the inequality:
\[ 4 \geq 4k \]
- Divide both sides by 4:
\[ 1 \geq k \]
- This can also be written as:
\[ k \leq 1 \]

### Conclusion:
The condition for the roots of the quadratic equation \(x^2 + 2x + k = 0\) to be real is:
\[ k \leq 1 \]