if 2 X square + 3 y square + 4 x square minus root 6 x y minus 2 root 3 y z minus 2 root 2 X is equals to zero then find the value of 2 X square + 3 y square + 16 Z square + 2 root 6 x y - 3 Y Z - 8 root 2 x z
Answers
Given : 2x² + 3y² + 4z² - √6xy - 2√3yz - 2√2xz = 0
To find : 2x² + 3y² + 16z² + 2√6xy - 8√3yz - 8√2xz
Solution:
2x² + 3y² + 4z² - √6xy - 2√3yz - 2√2xz = 0
=> 2(2x² + 3y² + 4z² - √6xy - 2√3yz - 2√2xz ) = 0
=> 2x² + 3y² - 2√6xy + 3y² + 4z² - 4√3yz + 2x² + 4z² - 4√2xz = 0
=> (√2x - √3y)² + (√3y - 2z)² + (√2x - 2z)² = 0
as each term is square and sum of all terms = 0
hence each term = 0
=> √2x - √3y = 0 , √3y - 2z = 0 , √2x - 2z = 0
=> √2x = √3y , √3y = 2z , √2x = 2z
=> √2x = √3y = 2z
=> 2x² = 3y² = 4z²
2x² + 3y² + 16z² + 2√6xy - 8√3yz - 8√2xz
= 2x² + 2x² + 4(2x²) + 2√2x√3y - 8(√2x )(√2x/2) - 8√2x (√2x /2)
= 4x² + 8x² + 2√2x√2x - 8x² - 8x²
= -4x² + 4x²
= 0
2x² + 3y² + 16z² + 2√6xy - 8√3yz - 8√2xz = 0
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