Find the value of k so that the equation x
2 − 10x + 2k = 8x − k has exactly one
root.

We are given the quadratic equation:

\[
x^2 - 10x + 2k = 8x - k
\]

First, rearrange the equation to standard quadratic form:

\[
x^2 - 10x + 2k - 8x + k = 0
\]

Simplify the terms:

\[
x^2 - 18x + 3k = 0
\]

The equation is now in the form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \),
- \( b = -18 \),
- \( c = 3k \).

For the quadratic equation to have exactly **one root**, the discriminant \( \Delta \) must be zero. The discriminant for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

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

Substitute \( a = 1 \), \( b = -18 \), and \( c = 3k \):

\[
\Delta = (-18)^2 - 4(1)(3k)
\]

\[
\Delta = 324 - 12k
\]

Set the discriminant equal to zero for exactly one root:

\[
324 - 12k = 0
\]

Solve for \( k \):

\[
12k = 324
\]

\[
k = \frac{324}{12} = 27
\]

Thus, the value of \( k \) that makes the equation have exactly one root is \( \boxed{27} \).

---

Would you like to see more details about how discriminants work, or have any other questions?

Here are five related questions:
1. What happens when the discriminant is greater than zero?
2. How does changing the value of \( k \) affect the number of roots?
3. Can you show an example of a quadratic equation with two distinct roots?
4. What is the significance of the vertex of a quadratic curve in relation to its roots?
5. How would you solve this equation if \( k = 10 \)?

**Tip:** For quadratic equations, if the discriminant is positive, there are two distinct roots. If it is zero, there is exactly one root, and if negative, no real roots.

Find the value of k so that the equation x^2 − 10x + 2k = 8x − k has exactly one root.

This detailed guide walks through solving the quadratic equation x^2 − 10x + 2k = 8x − k, focusing on finding the value of k that ensures exactly one root. It includes key algebraic concepts like discriminants and quadratic equations. Suitable for students in Grades 9-11.

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