$$ \frac{x}{y}+\frac{y}{x}=\frac{17}{4} $$, calculate two values of $$ \frac{x}{y} $$

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Let's denote $$ z = \frac{x}{y} $$. Then, we can rewrite the equation as:

$$
z + \frac{1}{z} = \frac{17}{4}
$$

To eliminate the fraction, we can multiply both sides by $$ 4z $$:

$$
4z^2 + 4 = 17z
$$

Rearranging this gives us a quadratic equation:

$$
4z^2 - 17z + 4 = 0
$$

Now, we will solve this quadratic equation for $$ z $$. The quadratic formula is given by:

$$
z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

In our case, $$ a = 4 $$, $$ b = -17 $$, and $$ c = 4 $$. Let's calculate the values of $$ z $$.

First, we will compute the discriminant $$ b^2 - 4ac $$:

$$
b^2 - 4ac = (-17)^2 - 4 \cdot 4 \cdot 4
$$

Now, let's calculate this step by step.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-17\right)^{2}-4\times 4\times 4$$
- step1: Rewrite the expression:
  $$\left(-17\right)^{2}-4^{3}$$
- step2: Rewrite the expression:
  $$17^{2}-4^{3}$$
- step3: Evaluate the power:
  $$289-4^{3}$$
- step4: Evaluate the power:
  $$289-64$$
- step5: Subtract the numbers:
  $$225$$
The discriminant is calculated as follows:

$$
b^2 - 4ac = 225
$$

Now, we can use the quadratic formula to find the values of $$ z $$:

$$
z = \frac{17 \pm \sqrt{225}}{2 \cdot 4}
$$

Calculating $$ \sqrt{225} $$:

$$
\sqrt{225} = 15
$$

Now substituting this back into the formula gives us:

$$
z = \frac{17 \pm 15}{8}
$$

This results in two possible values for $$ z $$:

1. $$ z_1 = \frac{17 + 15}{8} $$
2. $$ z_2 = \frac{17 - 15}{8} $$

Let's calculate these two values.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(17+15\right)}{8}$$
- step1: Remove the parentheses:
  $$\frac{17+15}{8}$$
- step2: Add the numbers:
  $$\frac{32}{8}$$
- step3: Reduce the numbers:
  $$\frac{4}{1}$$
- step4: Calculate:
  $$4$$
Calculate or simplify the expression $$ (17 - 15) / 8 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(17-15\right)}{8}$$
- step1: Remove the parentheses:
  $$\frac{17-15}{8}$$
- step2: Subtract the numbers:
  $$\frac{2}{8}$$
- step3: Reduce the fraction:
  $$\frac{1}{4}$$
The two values of $$ z = \frac{x}{y} $$ are:

1. $$ z_1 = 4 $$
2. $$ z_2 = \frac{1}{4} $$ (or $$ 0.25 $$)

Thus, the two values of $$ \frac{x}{y} $$ are $$ 4 $$ and $$ \frac{1}{4} $$.
The two values of $$ \frac{x}{y} $$ are 4 and $$ \frac{1}{4} $$.

, calculate two values of

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