Let's solve and simplify the given expressions step by step.
1. First Expression:
We need to multiply and simplify:
[tex]\[
(\sqrt{x} - 2\sqrt{2})(\sqrt{x} + 2\sqrt{2})
\][/tex]
This expression is in the form of a difference of squares:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
where [tex]\( a = \sqrt{x} \)[/tex] and [tex]\( b = 2\sqrt{2} \)[/tex].
Applying the formula:
[tex]\[
(\sqrt{x} - 2\sqrt{2})(\sqrt{x} + 2\sqrt{2}) = (\sqrt{x})^2 - (2\sqrt{2})^2
\][/tex]
Simplify each term:
[tex]\[
(\sqrt{x})^2 = x
\][/tex]
[tex]\[
(2\sqrt{2})^2 = 4 \cdot 2 = 8
\][/tex]
So the expression simplifies to:
[tex]\[
x - 8
\][/tex]
2. Second Expression:
Next, we need to simplify:
[tex]\[
(\sqrt{x} - \sqrt{2})^2
\][/tex]
This is a binomial squared, which can be expanded using the formula:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
where [tex]\( a = \sqrt{x} \)[/tex] and [tex]\( b = \sqrt{2} \)[/tex].
Applying the formula:
[tex]\[
(\sqrt{x} - \sqrt{2})^2 = (\sqrt{x})^2 - 2(\sqrt{x})(\sqrt{2}) + (\sqrt{2})^2
\][/tex]
Simplify each term:
[tex]\[
(\sqrt{x})^2 = x
\][/tex]
[tex]\[
(\sqrt{2})^2 = 2
\][/tex]
[tex]\[
2(\sqrt{x})(\sqrt{2}) = 2\sqrt{2x}
\][/tex]
Combining these, we get:
[tex]\[
(\sqrt{x} - \sqrt{2})^2 = x - 2\sqrt{2x} + 2
\][/tex]
Therefore, the simplified forms of the expressions are:
[tex]\[
(\sqrt{x} - 2\sqrt{2})(\sqrt{x} + 2\sqrt{2}) = x - 8
\][/tex]
and
[tex]\[
(\sqrt{x} - \sqrt{2})^2 = x - 2\sqrt{2x} + 2
\][/tex]
Thus, the filled square in the final result box would be:
[tex]\[
(\sqrt{x}-\sqrt{2})^2 = x - 2\sqrt{2x} + 2
\][/tex]
