To simplify the expression, [tex]\(2x^2 + y^2 + 2z^2 - 2\sqrt{2}xy + 2\sqrt{2}yz - 4xz\)[/tex], let’s follow these steps systematically:
1. Identify and group terms:
The expression contains several quadratic and mixed terms. We first list them:
[tex]\[
2x^2 + y^2 + 2z^2 - 2\sqrt{2}xy + 2\sqrt{2}yz - 4xz
\][/tex]
2. Combine like terms:
Although the terms [tex]\(2x^2, y^2, 2z^2\)[/tex] are already simplified, the mixed terms [tex]\( -2\sqrt{2}xy, 2\sqrt{2}yz, -4xz\)[/tex] need to be considered for further simplification.
3. Examine terms with similar factors:
Notice how the mixed terms involve combinations of [tex]\(x, y,\)[/tex] and [tex]\(z\)[/tex]. Let's handle them carefully:
[tex]\[
2x^2 - 2\sqrt{2}xy - 4xz + y^2 + 2\sqrt{2}yz + 2z^2
\][/tex]
4. Rewrite the expression:
No additional combination directly simplifies these mixed terms. This means the expression doesn't reduce further by conventional algebraic methods without additional context.
Ultimately, the expression [tex]\(2x^2 + y^2 + 2z^2 - 2\sqrt{2}xy + 2\sqrt{2}yz - 4xz\)[/tex] simplifies to:
[tex]\[
2x^2 - 2\sqrt{2}xy - 4xz + y^2 + 2\sqrt{2}yz + 2z^2
\][/tex]
This step-by-step approach clarifies how the initial terms combine directly into the final simplified form.
