To solve the expression [tex]\((1 + \sqrt{3})^2\)[/tex] and express it in the form [tex]\(a + b\sqrt{3}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers, we follow these steps:
1. Expand the Expression:
Given expression: [tex]\((1 + \sqrt{3})^2\)[/tex]
To expand the expression, we use the formula [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex].
Here, [tex]\(a = 1\)[/tex] and [tex]\(b = \sqrt{3}\)[/tex].
2. Calculate Each Component:
- [tex]\(a^2\)[/tex]:
[tex]\[
1^2 = 1
\][/tex]
- [tex]\(2ab\)[/tex]:
[tex]\[
2 \cdot 1 \cdot \sqrt{3} = 2\sqrt{3}
\][/tex]
- [tex]\(b^2\)[/tex]:
[tex]\[
(\sqrt{3})^2 = 3
\][/tex]
3. Combine the Results:
Add the computed values:
[tex]\[
a^2 + 2ab + b^2 = 1 + 2\sqrt{3} + 3
\][/tex]
Grouping the rational and irrational parts:
[tex]\[
(1 + 3) + 2\sqrt{3} = 4 + 2\sqrt{3}
\][/tex]
4. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
From the combined result, we can see that:
[tex]\(a = 4\)[/tex] (the coefficient of the rational part)
[tex]\(b = 2\)[/tex] (the coefficient of [tex]\(\sqrt{3}\)[/tex])
Therefore, the values are:
[tex]\[
a = 4
\][/tex]
[tex]\[
b = 2
\][/tex]
