Alright, to express the function [tex]\( f(x) = (x + \sqrt{2})^2 + (3x - 5\sqrt{8})^2 \)[/tex] in the form [tex]\( ax^2 + bx\sqrt{2} + c \)[/tex], we need to expand each term and then combine them.
1. Expanding [tex]\((x + \sqrt{2})^2\)[/tex]:
Let's start with the first term [tex]\((x + \sqrt{2})^2\)[/tex].
[tex]\[
(x + \sqrt{2})^2 = x^2 + 2x\sqrt{2} + (\sqrt{2})^2
\][/tex]
Since [tex]\((\sqrt{2})^2 = 2\)[/tex], we have:
[tex]\[
(x + \sqrt{2})^2 = x^2 + 2x\sqrt{2} + 2
\][/tex]
2. Expanding [tex]\((3x - 5\sqrt{8})^2\)[/tex]:
Next, let's expand the second term [tex]\( (3x - 5\sqrt{8})^2 \)[/tex].
First, simplify [tex]\(5\sqrt{8}\)[/tex]. Note that [tex]\(\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}\)[/tex], so:
[tex]\[
5\sqrt{8} = 5 \cdot 2\sqrt{2} = 10\sqrt{2}
\][/tex]
Now, expand [tex]\( (3x - 10\sqrt{2})^2 \)[/tex]:
[tex]\[
(3x - 10\sqrt{2})^2 = (3x)^2 - 2 \cdot (3x) \cdot (10\sqrt{2}) + (10\sqrt{2})^2
\][/tex]
[tex]\[
= 9x^2 - 60x\sqrt{2} + 100 \cdot 2
\][/tex]
[tex]\[
= 9x^2 - 60x\sqrt{2} + 200
\][/tex]
3. Combining the expanded terms:
Now, we combine the expanded forms of the two terms:
[tex]\[
f(x) = (x^2 + 2x\sqrt{2} + 2) + (9x^2 - 60x\sqrt{2} + 200)
\][/tex]
Combine like terms:
[tex]\[
f(x) = x^2 + 9x^2 + 2x\sqrt{2} - 60x\sqrt{2} + 2 + 200
\][/tex]
Simplify:
[tex]\[
f(x) = 10x^2 - 58x\sqrt{2} + 202
\][/tex]
Thus, expressed in the form [tex]\( ax^2 + bx\sqrt{2} + c \)[/tex], we have:
[tex]\[ a = 10 \][/tex]
[tex]\[ b = -58 \][/tex]
[tex]\[ c = 202 \][/tex]
So, the final expression is:
[tex]\[
f(x) = 10x^2 - 58x\sqrt{2} + 202
\][/tex]
