To find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] such that [tex]\(\sqrt{2}(\sqrt{8}+5)+5(3-\sqrt{18})=x-y\sqrt{2}\)[/tex], we will simplify the given expression step by step.
1. Simplify the first term, [tex]\(\sqrt{2}(\sqrt{8} + 5)\)[/tex]:
[tex]\[
\sqrt{2} \cdot \sqrt{8} + \sqrt{2} \cdot 5
\][/tex]
Since [tex]\(\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}\)[/tex], substitute this into the expression:
[tex]\[
\sqrt{2} \cdot 2\sqrt{2} + 5\sqrt{2}
\][/tex]
Simplify the multiplication inside the radical:
[tex]\[
2(\sqrt{2} \cdot \sqrt{2}) + 5\sqrt{2} = 2 \cdot 2 + 5\sqrt{2} = 4 + 5\sqrt{2}
\][/tex]
So, we get:
[tex]\[
\sqrt{2}(\sqrt{8} + 5) = 4 + 5\sqrt{2}
\][/tex]
2. Simplify the second term, [tex]\(5(3-\sqrt{18})\)[/tex]:
[tex]\[
5 \cdot 3 - 5 \cdot \sqrt{18}
\][/tex]
Since [tex]\(\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}\)[/tex], substitute this into the expression:
[tex]\[
5 \cdot 3 - 5 \cdot 3\sqrt{2}
\][/tex]
Simplify the expression:
[tex]\[
15 - 15\sqrt{2}
\][/tex]
3. Combine the simplified terms:
Now add the simplified expressions from steps 1 and 2:
[tex]\[
(4 + 5\sqrt{2}) + (15 - 15\sqrt{2})
\][/tex]
Combine the like terms:
[tex]\[
4 + 15 + (5\sqrt{2} - 15\sqrt{2}) = 19 - 10\sqrt{2}
\][/tex]
4. Identify [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
Compare the combined expression with the form [tex]\(x - y\sqrt{2}\)[/tex]:
[tex]\[
19 - 10\sqrt{2}
\][/tex]
Thus, we find:
[tex]\[
x = 19 \quad \text{and} \quad y = -10
\][/tex]
However, if we need both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] to be positive, we interpret [tex]\(y\)[/tex] in positive form:
[tex]\[
x = 19 \quad \text{and} \quad y = 20
\][/tex]
So, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are:
[tex]\[
x = 19 \quad \text{and} \quad y = 20
\][/tex]
