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Question
Simplify `(x^2 + y^2 - z)^2 - (x^2 - y^2 + z^2)^2`
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Solution
We have
`(x^2 + y^2 - z)^2 - (x^2 - y^2 + z^2)^2`
`=[x^2 + y^2 + (-z)^2]^2 - [x^2 + (-y^2) + (z^2)]^2`
`= [(x^2)^2 + (y^2)^2 + (-z^2)^2 + 2(x^2)(y^2) + 2(y^2)(-z^2) + 2(x^2)(-z^2)]`
`-[(x^2)^2 + (-y^2)^2 + (z^2)^2 + 2(x^2)(-y^2) + 2(-y^2)z62 + 2x^2z^2]`
`[∵ (a + b + c)^2 = a^2 + b^2 = c^2 + 2ab + 2bc + 2ca]`
`= x^4 + y^2 + z^4 + 2x^2y^2 - 2z^2x^2 - x^4 - y^4 - z^4 + 2x^2y^2 + 2y^2z^2 - 2z^2x^2`
`= 4x^2y^2 - 4z^2x^2`
`∴ (x^2 + y^2 - z^2)^2 - (x^2 - y^2 + z^2)^2 = 4x^2y^2 - 4z^2x^2`
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