The equation
\[px^2-17x+6=0\] has two distinct roots,
\[\alpha\] and
\[\beta\].
Also,

\[\dfrac{1}{\alpha}-\dfrac{1}{\beta}=\dfrac{13}{6}\].
Find the value of
\[p\].

We are given the quadratic equation

\[
px^2 - 17x + 6 = 0
\]

with roots \(\alpha\) and \(\beta\). We are also given that

\[
\frac{1}{\alpha} - \frac{1}{\beta} = \frac{13}{6}.
\]

First, we know from Vieta's formulas that the sum and product of the roots \(\alpha\) and \(\beta\) for the quadratic equation \(ax^2 + bx + c = 0\) are given by:

\[
\alpha + \beta = -\frac{b}{a}
\]
\[
\alpha \beta = \frac{c}{a}.
\]

For our specific equation \(px^2 - 17x + 6 = 0\), we have:

\[
\alpha + \beta = \frac{17}{p}
\]
\[
\alpha \beta = \frac{6}{p}.
\]

Next, let's work with the given expression:

\[
\frac{1}{\alpha} - \frac{1}{\beta} = \frac{\beta - \alpha}{\alpha \beta}.
\]

Substituting the expressions for \(\alpha + \beta\) and \(\alpha \beta\), we have:

\[
\frac{\beta - \alpha}{\alpha \beta} = \frac{13}{6}.
\]

Since \(\beta - \alpha\) can be expressed as:

\[
\beta - \alpha = \sqrt{(\alpha + \beta)^2 - 4\alpha \beta},
\]

we substitute:

\[
\frac{\sqrt{\left(\frac{17}{p}\right)^2 - 4 \cdot \frac{6}{p}}}{\frac{6}{p}} = \frac{13}{6}.
\]

Simplifying further:

\[
\frac{\sqrt{\frac{289}{p^2} - \frac{24}{p}}}{\frac{6}{p}} = \frac{13}{6}.
\]

This simplifies to:

\[
\frac{\sqrt{\frac{289 - 24p}{p^2}}}{\frac{6}{p}} = \frac{13}{6}.
\]

Simplifying further:

\[
\frac{\sqrt{289 - 24p}}{6} = \frac{13}{6}.
\]

This implies:

\[
\sqrt{289 - 24p} = 13.
\]

Squaring both sides:

\[
289 - 24p = 169.
\]

Solving for \(p\):

\[
24p = 289 - 169 = 120,
\]

\[
p = \frac{120}{24} = 5.
\]

Thus, the value of \(p\) is \( \boxed{5} \).

Would you like to explore the details further or have any additional questions?

### Related Questions:
1. What happens if \(p\) takes a different value in the quadratic equation?
2. How can Vieta's formulas be applied to higher-degree polynomials?
3. What is the significance of distinct roots in a quadratic equation?
4. How do the roots change if the discriminant of a quadratic equation is zero?
5. Can we solve the quadratic equation using a different method, like completing the square?
6. How does the value of \(p\) affect the nature of the roots (real, complex)?
7. What would be the impact of a negative value of \(p\) on the roots?
8. How can we derive the quadratic formula from the general form of a quadratic equation?

### Tip:
Always check if the discriminant of the quadratic equation is non-negative to ensure the roots are real when dealing with real numbers.

Learn how to solve the quadratic equation px^2 - 17x + 6 = 0 with distinct roots α and β, where 1/α - 1/β = 13/6. This problem utilizes Vieta's formulas and explores the relationship between roots and coefficients. Suitable for advanced high school students and college-level algebra courses.

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