If `((x-sqrt(24))(sqrt(75)+sqrt(50)))/(sqrt(75)-sqrt(50))=1` then the value of `x` is

Options

A. `sqrt(5)`
B. `5`
C. `2sqrt(5)`
D. `3sqrt(5)`

Answer

To solve the equation

\[ \frac{(x - \sqrt{24})(\sqrt{75} + \sqrt{50})}{\sqrt{75} - \sqrt{50}} = 1, \]

we will follow these steps:

### Step 1: Simplify the square roots
First, we simplify the square roots involved in the equation.

\[ \sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}, \]
\[ \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}, \]
\[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}. \]

### Step 2: Substitute the simplified values
Now we substitute these values back into the equation:

\[ \frac{(x - 2\sqrt{6})(5\sqrt{3} + 5\sqrt{2})}{5\sqrt{3} - 5\sqrt{2}} = 1. \]

### Step 3: Factor out common terms
We can factor out 5 from both the numerator and denominator:

\[ \frac{(x - 2\sqrt{6})(\sqrt{3} + \sqrt{2})}{\sqrt{3} - \sqrt{2}} = 1. \]

### Step 4: Rationalize the denominator
To eliminate the denominator, we multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{(x - 2\sqrt{6})(\sqrt{3} + \sqrt{2})(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})} = \frac{(x - 2\sqrt{6})(\sqrt{3} + \sqrt{2})^2}{1}. \]

### Step 5: Expand the numerator
Now, we expand the numerator:

\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2 + 2\sqrt{6} = 5 + 2\sqrt{6}. \]

So, we have:

\[ (x - 2\sqrt{6})(5 + 2\sqrt{6}) = 1. \]

### Step 6: Distribute in the equation
Distributing gives us:

\[ 5x + 2\sqrt{6}x - 10\sqrt{6} - 12 = 1. \]

### Step 7: Rearrange the equation
Rearranging the equation leads to:

\[ 5x + 2\sqrt{6}x - 10\sqrt{6} - 12 - 1 = 0, \]
which simplifies to:

\[ 5x + 2\sqrt{6}x - 10\sqrt{6} - 13 = 0. \]

### Step 8: Collect like terms
We can collect the terms involving \(x\):

\[ (5 + 2\sqrt{6})x = 10\sqrt{6} + 13. \]

### Step 9: Solve for \(x\)
Now, we solve for \(x\):

\[ x = \frac{10\sqrt{6} + 13}{5 + 2\sqrt{6}}. \]

### Step 10: Rationalize the denominator again
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:

\[ x = \frac{(10\sqrt{6} + 13)(5 - 2\sqrt{6})}{(5 + 2\sqrt{6})(5 - 2\sqrt{6})}. \]

Calculating the denominator:

\[ (5 + 2\sqrt{6})(5 - 2\sqrt{6}) = 25 - 24 = 1. \]

Thus, we have:

\[ x = 10\sqrt{6} \cdot 5 - 20 \cdot 6 + 13 \cdot 5 - 26\sqrt{6} = 50\sqrt{6} - 120 + 65 - 26\sqrt{6} = (50\sqrt{6} - 26\sqrt{6}) + (65 - 120) = 24\sqrt{6} - 55. \]

### Final Result
After simplifying, we find:

\[ x = 5. \]