`sqrt((x//y))+sqrt((y//x))=5//2, x +y= 6`, then x =

Options

A. `(1,5)`
B. `(3,3)`
C. `((6)/(5), (24)/(5))`
D. none

Answer

To solve the problem step by step, we start with the given equations:

1. \(\sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = \frac{5}{2}\)
2. \(x + y = 6\)

### Step 1: Simplify the first equation

We can rewrite the first equation as follows:

\[ \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = \frac{5}{2} \]

This can be expressed as:

\[ \frac{\sqrt{x}}{\sqrt{y}} + \frac{\sqrt{y}}{\sqrt{x}} = \frac{5}{2} \]

### Step 2: Find a common denominator

To combine the fractions on the left side, we find a common denominator:

\[ \frac{x + y}{\sqrt{xy}} = \frac{5}{2} \]

### Step 3: Substitute \(x + y\)

From the second equation, we know \(x + y = 6\). Substitute this into the equation:

\[ \frac{6}{\sqrt{xy}} = \frac{5}{2} \]

### Step 4: Cross-multiply to solve for \(\sqrt{xy}\)

Cross-multiplying gives:

\[ 6 \cdot 2 = 5 \sqrt{xy} \]

This simplifies to:

\[ 12 = 5 \sqrt{xy} \]

### Step 5: Solve for \(\sqrt{xy}\)

Dividing both sides by 5:

\[ \sqrt{xy} = \frac{12}{5} \]

### Step 6: Square both sides to find \(xy\)

Squaring both sides results in:

\[ xy = \left(\frac{12}{5}\right)^2 = \frac{144}{25} \]

### Step 7: Use the identity \(x^2 - (x+y)x + xy = 0\)

We can use the identity \(x^2 - (x+y)x + xy = 0\) to form a quadratic equation:

\[ x^2 - 6x + \frac{144}{25} = 0 \]

### Step 8: Multiply through by 25 to eliminate the fraction

Multiplying the entire equation by 25 gives:

\[ 25x^2 - 150x + 144 = 0 \]

### Step 9: Use the quadratic formula

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

Here, \(a = 25\), \(b = -150\), and \(c = 144\).

Calculating the discriminant:

\[ b^2 - 4ac = (-150)^2 - 4 \cdot 25 \cdot 144 = 22500 - 14400 = 8100 \]

Now, substituting into the quadratic formula:

\[ x = \frac{150 \pm \sqrt{8100}}{50} \]

### Step 10: Calculate \(\sqrt{8100}\)

Calculating \(\sqrt{8100}\):

\[ \sqrt{8100} = 90 \]

### Step 11: Find the values of \(x\)

Substituting back gives:

\[ x = \frac{150 \pm 90}{50} \]

Calculating the two possible values:

1. \(x = \frac{240}{50} = \frac{24}{5}\)
2. \(x = \frac{60}{50} = \frac{6}{5}\)

### Conclusion

Thus, the possible values of \(x\) are \(\frac{24}{5}\) and \(\frac{6}{5}\).