The quadratic equation is given as:

$x^2 + 8kx + k = 0$

For the quadratic equation $ax^2 + bx + c = 0$ to have equal roots, the discriminant must be zero. The discriminant $\Delta$ is given by:

$\Delta = b^2 - 4ac$

In this case:

• $a = 1$
• $b = 8k$
• $c = k$

Thus, the discriminant becomes:

$\Delta = (8k)^2 - 4(1)(k)$ $\Delta = 64k^2 - 4k$

Since the equation has equal roots, we set the discriminant equal to zero:

$64k^2 - 4k = 0$

Factor the equation:

$4k(16k - 1) = 0$

This gives two possible solutions:

1. $4k = 0$, which implies $k = 0$
2. $16k - 1 = 0$, which implies $k = \frac{1}{16}$

Thus, the values of $k$ are $k = 0$ or $k = \frac{1}{16}$.

Would you like more details on how the discriminant works, or do you have any other questions?

Here are 5 related questions:

1. What are the properties of a quadratic equation with equal roots?
2. How do you solve quadratic equations by factoring?
3. What does the discriminant tell us about the nature of the roots of a quadratic equation?
4. How do you find the roots of a quadratic equation using the quadratic formula?
5. How do you derive the condition for equal roots in a quadratic equation?

Tip: In quadratic equations, if the discriminant is positive, the roots are real and distinct; if it's zero, the roots are real and equal; and if it's negative, the roots are complex.