To simplify the expression [tex]\( 2x^2 + y^2 + 8z^2 - 2\sqrt{2}xy + 4\sqrt{2}yz - 8xz \)[/tex], follow these steps:
1. Rewrite Original Expression:
Start with the given expression:
[tex]\[
2x^2 + y^2 + 8z^2 - 2\sqrt{2}xy + 4\sqrt{2}yz - 8xz
\][/tex]
2. Group Like Terms:
Group terms that contain similar variable combinations and coefficients:
[tex]\[
2x^2 - 2\sqrt{2}xy - 8xz + y^2 + 4\sqrt{2}yz + 8z^2
\][/tex]
3. Examine Quadratic Nature:
Notice that this expression is a quadratic form in variables [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex].
4. Expand Each Term's Contribution:
Clearly write each squared term and cross-product term:
- [tex]\(2x^2\)[/tex]: The term with [tex]\(x^2\)[/tex].
- [tex]\(y^2\)[/tex]: The term with [tex]\(y^2\)[/tex].
- [tex]\(8z^2\)[/tex]: The term with [tex]\(z^2\)[/tex].
- [tex]\(-2\sqrt{2}xy\)[/tex]: The cross-product term between [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
- [tex]\(4\sqrt{2}yz\)[/tex]: The cross-product term between [tex]\(y\)[/tex] and [tex]\(z\)[/tex].
- [tex]\(-8xz\)[/tex]: The cross-product term between [tex]\(x\)[/tex] and [tex]\(z\)[/tex].
5. Combine with Identified Constants and Cross Terms:
Collect the terms and ensure they are correctly organized:
[tex]\[
2x^2 - 2\sqrt{2}xy - 8xz + y^2 + 4\sqrt{2}yz + 8z^2
\][/tex]
6. Final Expression:
Confirm that there are no further simplifications available. Each term is already simplified.
Therefore, the expression simplified and grouped appropriately is:
[tex]\[
2x^2 - 2\sqrt{2}xy - 8xz + y^2 + 4\sqrt{2}yz + 8z^2
\][/tex]
This conforms to the structured form for such quadratic expressions with mixed variable terms.
