Given that x2 + 8kx + k = 0 has equal roots, find the value of k.

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.

Given that x^2 + 8kx + k = 0 has equal roots, find the value of k.

Discover how to find the value of k in the quadratic equation x^2 + 8kx + k = 0, given that it has equal roots. This problem explores key algebraic concepts, including the discriminant formula and the condition for equal roots. Ideal for students in Grades 9-11.

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