Let `x^2-3x+p=0` has two positive roots a and b, the minimum value of `((4)/(a)+(1)/(b))` is ____________.

Explanation

To find the minimum value of \(\frac{4}{a} + \frac{1}{b}\) given that the quadratic equation \(x^2 - 3x + p = 0\) has two positive roots \(a\) and \(b\), we can use the properties of means.

### Step-by-step Solution:

1. **Identify the roots and their properties**:
The roots \(a\) and \(b\) of the quadratic equation can be expressed using Vieta's formulas:
\[ a + b = 3 \quad \text{(sum of roots)} \]
\[ ab = p \quad \text{(product of roots)} \]

2. **Express the function to minimize**:
We need to minimize the expression:
\[ \frac{4}{a} + \frac{1}{b} \]

3. **Use the AM-HM inequality**:
Since \(a\) and \(b\) are positive, we can apply the Arithmetic Mean - Harmonic Mean (AM-HM) inequality. For any positive numbers \(x_1, x_2, \ldots, x_n\):
\[ \text{AM} \geq \text{HM} \]
Specifically, for our case, we can consider:
\[ \frac{\frac{4}{a} + \frac{1}{b}}{2} \geq \frac{2}{\frac{a}{4} + \frac{b}{1}} \]

4. **Rearranging the inequality**:
We can rewrite the inequality:
\[ \frac{4}{a} + \frac{1}{b} \geq \frac{4 + 1}{\frac{a}{4} + b} = \frac{5}{\frac{a}{4} + b} \]

5. **Substituting \(b = 3 - a\)**:
From the sum of roots, we have \(b = 3 - a\). Substituting this into the expression gives:
\[ \frac{4}{a} + \frac{1}{3 - a} \]

6. **Finding the minimum value**:
To find the minimum of \(\frac{4}{a} + \frac{1}{3 - a}\), we can differentiate with respect to \(a\) and set the derivative to zero:
\[ f(a) = \frac{4}{a} + \frac{1}{3 - a} \]
\[ f'(a) = -\frac{4}{a^2} + \frac{1}{(3 - a)^2} \]
Setting \(f'(a) = 0\) gives:
\[ \frac{4}{a^2} = \frac{1}{(3 - a)^2} \]
Cross-multiplying and simplifying leads to:
\[ 4(3 - a)^2 = a^2 \]

7. **Solving the equation**:
Expanding and rearranging gives:
\[ 4(9 - 6a + a^2) = a^2 \]
\[ 36 - 24a + 4a^2 = a^2 \]
\[ 3a^2 - 24a + 36 = 0 \]
Dividing by 3:
\[ a^2 - 8a + 12 = 0 \]
Using the quadratic formula:
\[ a = \frac{8 \pm \sqrt{(8)^2 - 4 \cdot 1 \cdot 12}}{2 \cdot 1} = \frac{8 \pm \sqrt{64 - 48}}{2} = \frac{8 \pm \sqrt{16}}{2} = \frac{8 \pm 4}{2} \]
Thus, \(a = 6\) or \(a = 2\).

8. **Finding corresponding \(b\)**:
If \(a = 6\), then \(b = 3 - 6 = -3\) (not valid). If \(a = 2\), then \(b = 3 - 2 = 1\).

9. **Calculating the minimum value**:
Now substituting \(a = 2\) and \(b = 1\) into the original expression:
\[ \frac{4}{2} + \frac{1}{1} = 2 + 1 = 3 \]

Thus, the minimum value of \(\frac{4}{a} + \frac{1}{b}\) is \(\boxed{3}\).