Certainly! Let's solve the equation step by step:
[tex]\[
\sqrt{8}(\sqrt{2} - x) = 11
\][/tex]
1. Isolate the term involving [tex]\( x \)[/tex]:
Start by simplifying the equation. Notice that [tex]\(\sqrt{8}\)[/tex] can be rewritten in a simpler form:
[tex]\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}
\][/tex]
Substitute this back into the equation:
[tex]\[
2\sqrt{2}(\sqrt{2} - x) = 11
\][/tex]
2. Distribute [tex]\( 2\sqrt{2} \)[/tex]:
Expand the left side of the equation:
[tex]\[
2\sqrt{2} \cdot \sqrt{2} - 2\sqrt{2} \cdot x = 11
\][/tex]
Simplifying further:
[tex]\[
2 \cdot 2 - 2\sqrt{2} \cdot x = 11
\][/tex]
[tex]\[
4 - 2\sqrt{2} \cdot x = 11
\][/tex]
3. Isolate [tex]\( x \)[/tex]:
Subtract 4 from both sides of the equation:
[tex]\[
4 - 2\sqrt{2} \cdot x - 4 = 11 - 4
\][/tex]
[tex]\[
-2\sqrt{2} \cdot x = 7
\][/tex]
Next, divide both sides by [tex]\(-2\sqrt{2}\)[/tex]:
[tex]\[
x = \frac{7}{-2\sqrt{2}}
\][/tex]
4. Simplify the fraction:
To simplify [tex]\(\frac{7}{-2\sqrt{2}}\)[/tex], rationalize the denominator:
[tex]\[
x = \frac{7}{-2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{7\sqrt{2}}{-4}
\][/tex]
[tex]\[
x = -\frac{7\sqrt{2}}{4}
\][/tex]
5. Convert to decimal:
Finally, we can convert [tex]\(-\frac{7\sqrt{2}}{4}\)[/tex] to a decimal for a more precise answer. Using a calculator, we find that:
[tex]\[
x \approx -2.4748737341529163
\][/tex]
So, the solution to the equation [tex]\(\sqrt{8}(\sqrt{2} - x) = 11\)[/tex] is approximately:
[tex]\[
x \approx -2.4748737341529163
\][/tex]
