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The equation given is \( 2x^2 + 4x + k = 0 \), and we are asked to find the values of \( k \) for which the equation has real roots.

To determine whether a quadratic equation has real roots, we use the discriminant, which is given by the formula:

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

For the quadratic equation \( ax^2 + bx + c = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = 4 \)
- \( c = k \)

Substitute these values into the discriminant formula:

\[
\Delta = 4^2 - 4(2)(k) = 16 - 8k
\]

For the quadratic equation to have real roots, the discriminant must be greater than or equal to zero. So, we solve the inequality:

\[
16 - 8k \geq 0
\]

Solve for \( k \):

\[
16 \geq 8k
\]

\[
k \leq 2
\]

Therefore, the equation will have real roots if \( k \leq 2 \).

Would you like more details or have any questions about this?

Here are some related questions for further practice:
1. What happens to the nature of the roots if \( k > 2 \)?
2. How does the graph of \( 2x^2 + 4x + k = 0 \) change as \( k \) varies?
3. What are the roots of the equation when \( k = 2 \)?
4. How would the solution change if the coefficient of \( x^2 \) was different?
5. What are the conditions for the roots to be distinct or equal?

**Tip**: The discriminant helps determine not just the reality of the roots but also whether they are distinct or equal. \( \Delta > 0 \) means two distinct real roots, \( \Delta = 0 \) means a repeated real root, and \( \Delta < 0 \) implies complex roots.

Find the values of k for which the equation 2x^2 + 4x + k = 0 has real roots.

Learn how to find the values of k for which the quadratic equation 2x^2 + 4x + k = 0 has real roots. This problem explores the discriminant and its role in determining the nature of the roots for a quadratic equation. Suitable for students in Grades 9-10.

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