To solve the equation [tex]\( \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \)[/tex] and express it in the form [tex]\( a + b\sqrt{3} \)[/tex], follow these steps:
1. Rationalize the Denominator:
To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\( \sqrt{3} + 1 \)[/tex] is [tex]\( \sqrt{3} - 1 \)[/tex].
[tex]\[
\frac{\sqrt{3} - 1}{\sqrt{3} + 1} \cdot \frac{\sqrt{3} - 1}{\sqrt{3} - 1}
\][/tex]
2. Expand the Numerator:
Let's first multiply out the numerator:
[tex]\[
(\sqrt{3} - 1) \cdot (\sqrt{3} - 1) = (\sqrt{3})^2 - 2(\sqrt{3})(1) + (1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}
\][/tex]
3. Expand the Denominator:
Next, multiply out the denominator:
[tex]\[
(\sqrt{3} + 1) \cdot (\sqrt{3} - 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2
\][/tex]
4. Combine the Results:
Now, combine the results from steps 2 and 3:
[tex]\[
\frac{4 - 2\sqrt{3}}{2}
\][/tex]
5. Simplify the Fraction:
Simplify this fraction by dividing both the numerator terms by the denominator:
[tex]\[
\frac{4}{2} - \frac{2\sqrt{3}}{2} = 2 - \sqrt{3}
\][/tex]
6. Identify [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
Now, we can identify the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] from the simplified expression [tex]\( 2 - \sqrt{3} \)[/tex]:
[tex]\[
a = 0.2679491924311228, \qquad b = -1.7320508075688772
\][/tex]
So, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[ a = 0.2679491924311228 \][/tex]
[tex]\[ b = -1.7320508075688772 \][/tex]
