To solve the equation [tex]\(\frac{10 - \sqrt{18}}{\sqrt{2}} = a + b \sqrt{2}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers, follow these steps:
1. Rationalize the denominator:
[tex]\[
\frac{10 - \sqrt{18}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{(10 - \sqrt{18})\sqrt{2}}{(\sqrt{2})^2} = \frac{(10\sqrt{2} - \sqrt{18}\sqrt{2})}{2}
\][/tex]
2. Simplify the expression:
[tex]\[
\sqrt{18} = 3\sqrt{2}
\][/tex]
Substituting this back into the expression, we get:
[tex]\[
\frac{10\sqrt{2} - 3\sqrt{2}\sqrt{2}}{2} = \frac{10\sqrt{2} - 3 \cdot 2}{2} = \frac{10\sqrt{2} - 6}{2}
\][/tex]
3. Separate into two fractions:
[tex]\[
\frac{10\sqrt{2} - 6}{2} = \frac{10\sqrt{2}}{2} - \frac{6}{2}
\][/tex]
Simplify each fraction:
[tex]\[
\frac{10\sqrt{2}}{2} = 5\sqrt{2}
\][/tex]
[tex]\[
\frac{6}{2} = 3
\][/tex]
4. Combine the simplified fractions:
[tex]\[
5\sqrt{2} - 3
\][/tex]
Therefore, by comparing the simplified expression [tex]\(5\sqrt{2} - 3\)[/tex] to the original form [tex]\(a + b\sqrt{2}\)[/tex], we can identify that:
[tex]\[
a = -3 \quad \text{and} \quad b = 5
\][/tex]
So, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = -3, \quad b = 5
\][/tex]
