To solve for [tex]\(a^2 + b^2\)[/tex] given [tex]\(a = \frac{1}{2 + \sqrt{3}}\)[/tex] and [tex]\(b = \frac{1}{2 - \sqrt{3}}\)[/tex], we follow these steps:
1. Rationalize [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- For [tex]\(a\)[/tex]:
[tex]\[
a = \frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})}
\][/tex]
The denominator simplifies as follows:
[tex]\[
(2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1
\][/tex]
Therefore,
[tex]\[
a = 2 - \sqrt{3}
\][/tex]
- For [tex]\(b\)[/tex]:
[tex]\[
b = \frac{1}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = \frac{2 + \sqrt{3}}{(2 - \sqrt{3})(2 + \sqrt{3})}
\][/tex]
The denominator simplifies as follows:
[tex]\[
(2 - \sqrt{3})(2 + \sqrt{3}) = 4 - 3 = 1
\][/tex]
Therefore,
[tex]\[
b = 2 + \sqrt{3}
\][/tex]
2. Calculate [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
- For [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = (2 - \sqrt{3})^2 = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3}
\][/tex]
- For [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = (2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}
\][/tex]
3. Add [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
[tex]\[
a^2 + b^2 = (7 - 4\sqrt{3}) + (7 + 4\sqrt{3}) = 7 + 7 = 14
\][/tex]
Thus, the value of [tex]\(a^2 + b^2\)[/tex] is [tex]\(\boxed{14}\)[/tex].
