If `(4+3sqrt5)/(4-3sqrt5)=a+bsqrt5`,a, b are rational numbers, them (a, b)=

Options

A. `(61/29,(-24)/29)`
B. `((-61)/29,24/29)`
C. `(61/29,24/29)`
D. `((-61)/29,(-24)/29)`

Answer

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

### Step 1: Rationalize the denominator
To eliminate the square root in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(4 + 3\sqrt{5}\).

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

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

\[ (4 - 3\sqrt{5})(4 + 3\sqrt{5}) = 4^2 - (3\sqrt{5})^2 = 16 - 45 = -29 \]

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

\[ (4 + 3\sqrt{5})(4 + 3\sqrt{5}) = 4^2 + 2 \cdot 4 \cdot 3\sqrt{5} + (3\sqrt{5})^2 = 16 + 24\sqrt{5} + 45 = 61 + 24\sqrt{5} \]

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

\[ \frac{61 + 24\sqrt{5}}{-29} = \frac{61}{-29} + \frac{24\sqrt{5}}{-29} = -\frac{61}{29} - \frac{24}{29}\sqrt{5} \]

### Step 5: Identify \(a\) and \(b\)
From the expression \(-\frac{61}{29} - \frac{24}{29}\sqrt{5}\), we can identify:

\[ a = -\frac{61}{29}, \quad b = -\frac{24}{29} \]

Thus, the values of \(a\) and \(b\) are:

\[ (a, b) = \left(-\frac{61}{29}, -\frac{24}{29}\right) \]

### Final Answer
\[ (a, b) = \left(-\frac{61}{29}, -\frac{24}{29}\right) \]

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