The equation \[px^2-8x+4=0\] has two distinct roots.

Also, one of the roots is thrice the other.

**Find the value of \[p\].**

We are given the quadratic equation

\[
px^2 - 8x + 4 = 0
\]

with two distinct roots \(\alpha\) and \(\beta\), where one root is thrice the other. Let's assume \(\alpha = 3\beta\).

### Step 1: Apply Vieta's Formulas
From Vieta's formulas for the quadratic equation \(ax^2 + bx + c = 0\), the sum and product of the roots \(\alpha\) and \(\beta\) are given by:

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

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

For the equation \(px^2 - 8x + 4 = 0\), we have:

\[
\alpha + \beta = \frac{8}{p}
\]

\[
\alpha \beta = \frac{4}{p}.
\]

### Step 2: Use the Given Relationship Between Roots
Since \(\alpha = 3\beta\), substitute into the formulas:

1. **Sum of the roots:**

\[
\alpha + \beta = 3\beta + \beta = 4\beta.
\]

Equating with the Vieta's result:

\[
4\beta = \frac{8}{p}.
\]

This simplifies to:

\[
\beta = \frac{2}{p}.
\]

2. **Product of the roots:**

\[
\alpha \beta = 3\beta \cdot \beta = 3\beta^2.
\]

Equating with the Vieta's result:

\[
3\beta^2 = \frac{4}{p}.
\]

### Step 3: Solve for \(p\)
Substituting \(\beta = \frac{2}{p}\) into the product equation:

\[
3\left(\frac{2}{p}\right)^2 = \frac{4}{p}.
\]

Simplify and solve for \(p\):

\[
3 \cdot \frac{4}{p^2} = \frac{4}{p}.
\]

\[
\frac{12}{p^2} = \frac{4}{p}.
\]

Multiply both sides by \(p^2\) to clear the fraction:

\[
12 = 4p.
\]

Finally, solve for \(p\):

\[
p = \frac{12}{4} = 3.
\]

Thus, the value of \(p\) is \( \boxed{3} \).

Would you like more details on any part of the solution, or do you have further questions?

### Related Questions:
1. What would happen if the root condition (one root is thrice the other) changes?
2. Can we derive a general formula for \(p\) if the roots are related by any ratio?
3. How does the discriminant change when the roots have a specific relationship like this?
4. What if \(p\) is negative—how does it affect the roots?
5. How would the solution change if the equation had equal roots?
6. What is the geometric interpretation of roots in a quadratic equation?
7. Can we use the quadratic formula to verify the solution for \(p\)?
8. How does changing the coefficient of \(x^2\) affect the roots of a quadratic equation?

### Tip:
Always verify your solution by substituting the roots back into the original equation to ensure they satisfy the equation.

Discover how to find the value of p in the quadratic equation px^2 - 8x + 4 = 0, where one root is thrice the other. This problem involves applying Vieta's formulas and solving equations. Suitable for high school students studying quadratic equations and algebra.

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