Question
Factorize:
$2{x^2} + {y^2} + 8{z^2} - 2\sqrt 2 xy + 4\sqrt 2 yz - 8xz$
$2{x^2} + {y^2} + 8{z^2} - 2\sqrt 2 xy + 4\sqrt 2 yz - 8xz$
✓
Answer
$2{x^2} + {y^2} + 8{z^2} - 2\sqrt 2 xy + 4\sqrt 2 yz - 8xz$
We need to factorize the expression $2{x^2} + {y^2} + 8{z^2} - 2\sqrt 2 xy + 4\sqrt 2 yz - 8xz$
The expression $2{x^2} + {y^2} + 8{z^2} - 2\sqrt 2 xy + 4\sqrt 2 yz - 8xz$ can also be written as
${\left( { - \sqrt 2 x} \right)^2} + {\left( y \right)^2} + {\left( {2\sqrt 2 z} \right)^2} + 2 \times \left( { - \sqrt 2 x} \right) \times y + 2 \times y \times \left( {2\sqrt 2 z} \right) + 2 \times \left( {2\sqrt 2 z} \right) \times \left( { - \sqrt 2 x} \right).$
We can observe that, we can apply the identity ${\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca$ with respect to the expression
${\left( { - \sqrt 2 x} \right)^2} + {\left( y \right)^2} + {\left( {2\sqrt 2 z} \right)^2} + 2 \times \left( { - \sqrt 2 x} \right) \times y + 2 \times y \times \left( {2\sqrt 2 z} \right) + 2 \times \left( {2\sqrt 2 z} \right) \times \left( { - \sqrt 2 x} \right)$,to get
${\left( { - \sqrt 2 x + y + 2\sqrt 2 z} \right)^2}$
Therefore, we conclude that after factorizing the expression $2{x^2} + {y^2} + 8{z^2} - 2\sqrt 2 xy + 4\sqrt 2 yz - 8xz $ we get ${\left( { - \sqrt 2 x + y + 2\sqrt 2 z} \right)^2}$.
We need to factorize the expression $2{x^2} + {y^2} + 8{z^2} - 2\sqrt 2 xy + 4\sqrt 2 yz - 8xz$
The expression $2{x^2} + {y^2} + 8{z^2} - 2\sqrt 2 xy + 4\sqrt 2 yz - 8xz$ can also be written as
${\left( { - \sqrt 2 x} \right)^2} + {\left( y \right)^2} + {\left( {2\sqrt 2 z} \right)^2} + 2 \times \left( { - \sqrt 2 x} \right) \times y + 2 \times y \times \left( {2\sqrt 2 z} \right) + 2 \times \left( {2\sqrt 2 z} \right) \times \left( { - \sqrt 2 x} \right).$
We can observe that, we can apply the identity ${\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca$ with respect to the expression
${\left( { - \sqrt 2 x} \right)^2} + {\left( y \right)^2} + {\left( {2\sqrt 2 z} \right)^2} + 2 \times \left( { - \sqrt 2 x} \right) \times y + 2 \times y \times \left( {2\sqrt 2 z} \right) + 2 \times \left( {2\sqrt 2 z} \right) \times \left( { - \sqrt 2 x} \right)$,to get
${\left( { - \sqrt 2 x + y + 2\sqrt 2 z} \right)^2}$
Therefore, we conclude that after factorizing the expression $2{x^2} + {y^2} + 8{z^2} - 2\sqrt 2 xy + 4\sqrt 2 yz - 8xz $ we get ${\left( { - \sqrt 2 x + y + 2\sqrt 2 z} \right)^2}$.
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