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Question
Simplify the expression:
`(x + y + z)^2 + (x + y/2 + 2/3)^2 - (x/2 + y/3 + z/4)^2`
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Solution
We have,
`(x + y + z)^2 + (x + y/2 + 2/3)^2 - (x/2 + y/3 + z/4)^2`
`= [x^2 + y^2 + z^2 + 2xy + 2yz + 2zx] + [x^2 + y^2/4 + z^2/9 + 2x . y/2 + 2 (zx)/3 + (yz)/3] - [x^2/4 + y^2/9 + z^2/10 + (xy)/3 + (xz)/4 + (yz)/6]`
`= x^2 + y^2 + z^2 + x^2 + y^2/4 + z^2/9 - x^2/4 - y^2/9 - z^2/16 + 2xy + 2x . y/2 - (xy)/3 + 2yz + (yz)/3 - (yz)/6 + 2zx + (2zx)/3 - (xz)/4`
`= (8x^2 - x^2)/4 + (36y^2 + 9y^2 - 4y^2)/36 + (144z^2 + 16z^2 - 9z^2)/144 + (6xy + 3xy - xy)/3 + (13yz)/6 + (29xz)/12`
`= (7x^2)/4 + (41y^2)/36 + (151z^2)/144 + (8xy)/3 + (13yz)/6 + (29zx)/12`
`∴ (x + y + z)^2 + (x + y/2 + z/3)^2 - (x/2 + y/3 + z/4)^2`
`= (7x^2)/4 + (41y^2)/36 + (151Z^2)/144 + (8xy)/3 + (13yz)/6 + (29zx)/12`
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