For what value of k , `kx^(2) + 8x + 2 = 0` has real roots

Options

A. `k lt 8`
B. `k gt 8`
C. k = 8
D. none of these

Answer

To determine the value of \( k \) for which the quadratic equation \( kx^2 + 8x + 2 = 0 \) has real roots, we need to analyze the discriminant of the quadratic equation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = k \)
- \( b = 8 \)
- \( c = 2 \)

2. **Write the formula for the discriminant**:
The discriminant \( D \) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]

3. **Substitute the coefficients into the discriminant formula**:
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula, we get:
\[ D = 8^2 - 4(k)(2) \]
Simplifying this, we have:
\[ D = 64 - 8k \]

4. **Set the condition for real roots**:
For the quadratic equation to have real roots, the discriminant must be greater than or equal to zero:
\[ 64 - 8k \geq 0 \]

5. **Solve the inequality**:
Rearranging the inequality gives:
\[ 64 \geq 8k \]
Dividing both sides by 8, we obtain:
\[ 8 \geq k \]
or equivalently,
\[ k \leq 8 \]

6. **Conclusion**:
Therefore, the value of \( k \) must be less than or equal to 8 for the quadratic equation \( kx^2 + 8x + 2 = 0 \) to have real roots.

### Final Answer:
The value of \( k \) must be \( k \leq 8 \).