If both a and b are rational numbers find the values of a and b in the following equation . `(sqrt(3)-1)/(sqrt(3)+1)=a+bsqrt(3)`

Options

A. a=-1,b=2
B. a=1 , b=2
C. a=2, b=-1
D. a=-2,b=1

Answer

To solve the equation \(\frac{\sqrt{3}-1}{\sqrt{3}+1} = a + b\sqrt{3}\), where \(a\) and \(b\) are rational numbers, we will follow these steps:

### Step 1: Rationalize the Left Side
To simplify the left side, we can multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{3}-1\):

\[ \frac{(\sqrt{3}-1)(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)} \]

### Step 2: Expand the Numerator and Denominator
Now we will expand both the numerator and the denominator:

- **Numerator**:
\[ (\sqrt{3}-1)(\sqrt{3}-1) = (\sqrt{3})^2 - 2(\sqrt{3})(1) + (1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} \]

- **Denominator**:
\[ (\sqrt{3}+1)(\sqrt{3}-1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2 \]

### Step 3: Combine the Results
Now we can combine the results from the numerator and the denominator:

\[ \frac{4 - 2\sqrt{3}}{2} = \frac{4}{2} - \frac{2\sqrt{3}}{2} = 2 - \sqrt{3} \]

### Step 4: Compare with the Right Side
Now we have:

\[ 2 - \sqrt{3} = a + b\sqrt{3} \]

### Step 5: Identify the Values of \(a\) and \(b\)
To find the values of \(a\) and \(b\), we can compare the coefficients:

- The constant term on the left side is \(2\), which corresponds to \(a\).
- The coefficient of \(\sqrt{3}\) on the left side is \(-1\), which corresponds to \(b\).

Thus, we have:

\[ a = 2 \quad \text{and} \quad b = -1 \]

### Final Answer
The values of \(a\) and \(b\) are:
\[ a = 2, \quad b = -1 \]

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