If `(3+2sqrt(5))/(4-2sqrt(5))=p+qsqrt(5)` where p and q are rational numbers, then values of p and q respectively are

Options

A. `8,-7//2`
B. `-8,-7//2`
C. `+4,+7`
D. `-4,-7`

Answer

To solve the equation \(\frac{3 + 2\sqrt{5}}{4 - 2\sqrt{5}} = p + q\sqrt{5}\), where \(p\) and \(q\) are rational numbers, we will follow these steps:

### Step 1: Rationalize the denominator
To eliminate the square root from the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(4 + 2\sqrt{5}\).

\[ \frac{3 + 2\sqrt{5}}{4 - 2\sqrt{5}} \cdot \frac{4 + 2\sqrt{5}}{4 + 2\sqrt{5}} = \frac{(3 + 2\sqrt{5})(4 + 2\sqrt{5})}{(4 - 2\sqrt{5})(4 + 2\sqrt{5})} \]

### Step 2: Simplify the denominator
Using the difference of squares formula \(a^2 - b^2\), we calculate the denominator:

\[ (4 - 2\sqrt{5})(4 + 2\sqrt{5}) = 4^2 - (2\sqrt{5})^2 = 16 - 4 \cdot 5 = 16 - 20 = -4 \]

### Step 3: Expand the numerator
Now, we expand the numerator:

\[ (3 + 2\sqrt{5})(4 + 2\sqrt{5}) = 3 \cdot 4 + 3 \cdot 2\sqrt{5} + 2\sqrt{5} \cdot 4 + 2\sqrt{5} \cdot 2\sqrt{5} \]
\[ = 12 + 6\sqrt{5} + 8\sqrt{5} + 4 \cdot 5 \]
\[ = 12 + 14\sqrt{5} + 20 = 32 + 14\sqrt{5} \]

### Step 4: Combine the results
Now, we can combine the results from the numerator and denominator:

\[ \frac{32 + 14\sqrt{5}}{-4} = \frac{32}{-4} + \frac{14\sqrt{5}}{-4} = -8 - \frac{7}{2}\sqrt{5} \]

### Step 5: Identify \(p\) and \(q\)
From the expression \(-8 - \frac{7}{2}\sqrt{5}\), we can identify:

\[ p = -8, \quad q = -\frac{7}{2} \]

### Final Answer
The values of \(p\) and \(q\) respectively are:
\[ p = -8, \quad q = -\frac{7}{2} \]

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