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Let's solve question 18 step by step.

The given quadratic equation is:

\[
x^2 - kx + (k - 1) = 0
\]

Let \(\alpha\) and \(\beta\) be the roots of this quadratic equation. From Vieta's formulas, we know the following relationships between the roots \(\alpha\) and \(\beta\) and the coefficients of the quadratic equation:

1. **Sum of roots:** \(\alpha + \beta = k\)
2. **Product of roots:** \(\alpha \beta = k - 1\)

### Part a) Express \(\alpha\) and \(\beta\) in terms of \(k\)

We can use these Vieta relations to express \(\alpha\) and \(\beta\):

- \(\alpha + \beta = k\)
- \(\alpha \beta = k - 1\)

Thus, \(\alpha\) and \(\beta\) are determined by these two relations.

### Part b) Given that \(\alpha^2 + \beta^2 = 17\), find the possible values of \(k\)

We can use the identity:

\[
\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta
\]

Substitute \(\alpha + \beta = k\) and \(\alpha \beta = k - 1\) into this equation:

\[
\alpha^2 + \beta^2 = k^2 - 2(k - 1)
\]

Given that \(\alpha^2 + \beta^2 = 17\), we now have the equation:

\[
k^2 - 2(k - 1) = 17
\]

Simplify this equation:

\[
k^2 - 2k + 2 = 17
\]

\[
k^2 - 2k - 15 = 0
\]

Now solve this quadratic equation using the quadratic formula:

\[
k = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-15)}}{2(1)}
\]

\[
k = \frac{2 \pm \sqrt{4 + 60}}{2}
\]

\[
k = \frac{2 \pm \sqrt{64}}{2}
\]

\[
k = \frac{2 \pm 8}{2}
\]

Thus, the possible values of \(k\) are:

\[
k = \frac{2 + 8}{2} = 5 \quad \text{or} \quad k = \frac{2 - 8}{2} = -3
\]

### Final Answer:
The possible values of \(k\) are \(5\) and \(-3\).

---

Let me know if you would like further details on any step, and here are some related questions for further practice:

1. Solve a quadratic equation using Vieta’s formulas.
2. Find the roots of a quadratic equation when given specific relationships between the roots.
3. How can you derive identities like \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta\)?
4. What other properties can be derived from Vieta’s formulas?
5. Solve the equation \(k^2 - 2k - 15 = 0\) by factoring instead of using the quadratic formula.

**Tip:** Always double-check your results by substituting the values back into the original equations to ensure consistency.

Let α and β denote the roots of the quadratic equation x^2 - kx + (k - 1) = 0.
 a) Express α and β in terms of the real parameter k.
 b) Given that α^2 + β^2 = 17, find the possible values of k.

Learn how to express the roots of the quadratic equation x^2 - kx + (k - 1) = 0 in terms of k and solve for k given α^2 + β^2 = 17. This problem explores Vieta's formulas and quadratic identities, suitable for students in Grades 10-12.

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