If `sqrt(((x)/(1-x)))+sqrt(((1-x)/(x)))=2(1)/(6)`, then x is equal to

Options

A. `(3)/(13), (5)/(13)`
B. `(7)/(5), (3)/(5)`
C. `(9)/(13), (4)/(13)`
D. none

Answer

To solve the equation

\[ \sqrt{\frac{x}{1-x}} + \sqrt{\frac{1-x}{x}} = \frac{13}{6}, \]

we will follow these steps:

### Step 1: Simplify the equation
Let \( y = \sqrt{\frac{x}{1-x}} \). Then, we have

\[ \sqrt{\frac{1-x}{x}} = \frac{1}{y}. \]

Substituting this into the equation gives:

\[ y + \frac{1}{y} = \frac{13}{6}. \]

### Step 2: Multiply through by \( y \)
To eliminate the fraction, multiply both sides by \( y \):

\[ y^2 + 1 = \frac{13}{6}y. \]

### Step 3: Rearrange the equation
Rearranging gives us a standard quadratic equation:

\[ y^2 - \frac{13}{6}y + 1 = 0. \]

### Step 4: Multiply through by 6
To eliminate the fraction, multiply the entire equation by 6:

\[ 6y^2 - 13y + 6 = 0. \]

### Step 5: Apply the quadratic formula
Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 6, b = -13, c = 6 \):

\[ b^2 - 4ac = (-13)^2 - 4 \cdot 6 \cdot 6 = 169 - 144 = 25. \]

Thus,

\[ y = \frac{13 \pm \sqrt{25}}{12} = \frac{13 \pm 5}{12}. \]

### Step 6: Calculate the two possible values for \( y \)
Calculating gives:

1. \( y = \frac{18}{12} = \frac{3}{2} \)
2. \( y = \frac{8}{12} = \frac{2}{3} \)

### Step 7: Solve for \( x \)
Recall that \( y = \sqrt{\frac{x}{1-x}} \). We will solve for \( x \) for both values of \( y \).

#### For \( y = \frac{3}{2} \):

\[ \frac{3}{2} = \sqrt{\frac{x}{1-x}} \implies \left(\frac{3}{2}\right)^2 = \frac{x}{1-x} \implies \frac{9}{4} = \frac{x}{1-x}. \]

Cross-multiplying gives:

\[ 9(1-x) = 4x \implies 9 - 9x = 4x \implies 9 = 13x \implies x = \frac{9}{13}. \]

#### For \( y = \frac{2}{3} \):

\[ \frac{2}{3} = \sqrt{\frac{x}{1-x}} \implies \left(\frac{2}{3}\right)^2 = \frac{x}{1-x} \implies \frac{4}{9} = \frac{x}{1-x}. \]

Cross-multiplying gives:

\[ 4(1-x) = 9x \implies 4 - 4x = 9x \implies 4 = 13x \implies x = \frac{4}{13}. \]

### Final Answer
Thus, the possible values for \( x \) are:

\[ x = \frac{9}{13} \quad \text{or} \quad x = \frac{4}{13}. \]