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Finding Square of a Number Using Identity

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Question

Simplify : $(x^{2} + y^{2} - z^{2})^{2} - (x^{2} - y^{2} + z^{2})^{2}$

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Solution

Apply the formula $a^{2} - b^{2} = (a + b) (a - b)$ to simplify the expression.

${(x^{2} + y^{2} - z^{2})}^{2} - {(x^{2} - y^{2} + z^{2})}^{2}$

$= (x^{2} + y^{2} - z^{2} + x^{2} - y^{2} + z^{2}) {x^{2} + y^{2} - z^{2} - (x^{2} - y^{2} + z^{2})}$

$= (2 x^{2}) (x^{2} + y^{2} - z^{2} - x^{2} + y^{2} - z^{2})$

$= (2 x^{2}) (2 y^{2} - 2 z^{2})$

$= (4 x^{2}) (y^{2} - z^{2})$

$= 4 x^{2} y^{2} - 4 x^{2} z^{2}$

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