Roots of 3x^2+17x+16=0 are in the form of p:q, then find √p/√q + √q/√p

We are given the quadratic equation:

\[
3x^2 + 17x + 16 = 0
\]

### Step 1: Find the roots of the quadratic equation
To find the roots, we use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 3 \), \( b = 17 \), and \( c = 16 \).

Substitute these values into the quadratic formula:

\[
x = \frac{-17 \pm \sqrt{17^2 - 4(3)(16)}}{2(3)}
\]

\[
x = \frac{-17 \pm \sqrt{289 - 192}}{6}
\]

\[
x = \frac{-17 \pm \sqrt{97}}{6}
\]

So, the roots are:

\[
x_1 = \frac{-17 + \sqrt{97}}{6} \quad \text{and} \quad x_2 = \frac{-17 - \sqrt{97}}{6}
\]

### Step 2: Express the roots in the form \( p:q \)
Assume the roots are in the ratio \( p:q \). Let \( p = \frac{-17 + \sqrt{97}}{6} \) and \( q = \frac{-17 - \sqrt{97}}{6} \).

### Step 3: Find the expression \( \frac{\sqrt{p}}{\sqrt{q}} + \frac{\sqrt{q}}{\sqrt{p}} \)
The expression simplifies to:

\[
\frac{\sqrt{p}}{\sqrt{q}} + \frac{\sqrt{q}}{\sqrt{p}} = \sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}}
\]

### Step 4: Calculate \( \frac{p}{q} \)
Let's compute the ratio \( \frac{p}{q} \):

\[
\frac{p}{q} = \frac{\frac{-17 + \sqrt{97}}{6}}{\frac{-17 - \sqrt{97}}{6}} = \frac{-17 + \sqrt{97}}{-17 - \sqrt{97}}
\]

This can be simplified by multiplying the numerator and denominator by the conjugate of the denominator:

\[
\frac{p}{q} = \frac{(-17 + \sqrt{97})(-17 + \sqrt{97})}{(-17 - \sqrt{97})(-17 + \sqrt{97})}
\]

\[
= \frac{(-17 + \sqrt{97})^2}{(-17)^2 - (\sqrt{97})^2}
\]

\[
= \frac{289 - 2(17)\sqrt{97} + 97}{289 - 97} = \frac{386 - 34\sqrt{97}}{192}
\]

### Step 5: Simplify the expression
The expression \( \sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} \) simplifies to:

\[
\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} = \sqrt{k} + \frac{1}{\sqrt{k}} = k + \frac{1}{k} 
\]

But since both \( p \) and \( q \) are roots of the same quadratic equation, we know that \( \frac{p}{q} \) and \( \frac{q}{p} \) are reciprocals, leading to:

\[
\frac{\sqrt{p}}{\sqrt{q}} + \frac{\sqrt{q}}{\sqrt{p}} = 2
\]

### Final Answer:
The value of \( \frac{\sqrt{p}}{\sqrt{q}} + \frac{\sqrt{q}}{\sqrt{p}} \) is:

\[
\boxed{2}
\]

Would you like more details on any of the steps, or do you have any other questions?

### Related Questions:
1. How do you simplify the ratio of two roots given by a quadratic equation?
2. What is the significance of the expression \( \sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} \) when the roots are reciprocals?
3. How do you find the exact value of roots using the quadratic formula for different coefficients?
4. How does the discriminant affect the nature of roots in a quadratic equation?
5. What are the geometric interpretations of the roots of a quadratic equation?

### Tip:
When dealing with quadratic equations, always check if the roots can be expressed in a simple form or if they involve conjugates, as this often simplifies the calculations.

Learn how to find √p/√q + √q/√p for the roots of the quadratic equation 3x^2 + 17x + 16 = 0. This problem involves solving quadratic equations, determining ratios of roots, and simplifying expressions involving square roots. Suitable for advanced high school students and beyond.

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