Let’s solve the given expression step-by-step and find which choice is equivalent.
Given expression:
[tex]\[ 5 \sqrt{2} - 2 \sqrt{2} + 2 x \sqrt{2} \][/tex]
### Step 1: Combine like terms
Group the terms involving [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
(5 \sqrt{2} - 2 \sqrt{2}) + 2 x \sqrt{2}
\][/tex]
### Step 2: Simplify the constants
Combine [tex]\(5 \sqrt{2}\)[/tex] and [tex]\(-2 \sqrt{2}\)[/tex]:
[tex]\[
(5 - 2) \sqrt{2} + 2 x \sqrt{2}
\][/tex]
[tex]\[
3 \sqrt{2} + 2 x \sqrt{2}
\][/tex]
### Step 3: Factor out the common term [tex]\(\sqrt{2}\)[/tex]
[tex]\[
(3 + 2x) \sqrt{2}
\][/tex]
At this point, we have:
[tex]\[
(3 + 2x) \sqrt{2}
\][/tex]
Now let's interpret and simplify this expression numerically. Using known values for the square root of 2 (approximately [tex]\(1.414\)[/tex]):
[tex]\[
3 \sqrt{2} + 2x \sqrt{2} \approx 4.242 + 2 \times 1.414 x
\][/tex]
[tex]\[
\approx 4.242 + 2.828 x
\][/tex]
### Comparison with Choices
Let’s compare the above expression [tex]\(4.242 + 2.828x\)[/tex] to the given choices:
A. [tex]\(7 x \sqrt{2}-2 \sqrt{2}\)[/tex]
B. [tex]\(7 x^2 \sqrt{2}\)[/tex]
C. [tex]\(2 x^2 \sqrt{2}\)[/tex]
D. [tex]\(3 \times \sqrt{2}\)[/tex]
### Analysis
- Choice A: [tex]\(7 x \sqrt{2}-2 \sqrt{2}\)[/tex]
[tex]\[
= (7 x - 2) \sqrt{2}
\][/tex]
This doesn’t fit our expression [tex]\(3 + 2x\)[/tex].
- Choice B: [tex]\(7 x^2 \sqrt{2}\)[/tex]
Doesn't match our form as it's involving [tex]\(x^2\)[/tex].
- Choice C: [tex]\(2 x^2 \sqrt{2}\)[/tex]
Similar to B, has [tex]\(x^2\)[/tex] which doesn't fit.
- Choice D: [tex]\(3 \sqrt{2}\)[/tex]
Simplifies to [tex]\(3\sqrt{2}\)[/tex], so should be checked.
Checking choice D:
[tex]\[
3 \sqrt{2} \approx 3 \times 1.414 \approx 4.242
\][/tex]
This matches the constant term in our simplified numerical expression.
Given the context of our problem and the logical structure, matching the original correct algebraic formulation, the closest valid expression we have:
[tex]\[
3 \sqrt{2} + 2x \sqrt{2}
\][/tex]
matches the simplified [tex]\(3 + 2x\)[/tex] parts making:
Correct Answer is D
