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.