The two squares identity is (c) (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2
How to determine the identity?
The complete question is in the attached image
The identity is given as:
Two Squares identity
The Two Squares identity states that:
The difference of two squares is the product of the sum of the square roots and their difference
This means that
(a^2 + b^2)(c^2 + d^2) = a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2
Rewrite as:
(a^2 + b^2)(c^2 + d^2) = a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2
Add 0 to the equation
(a^2 + b^2)(c^2 + d^2) = a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2 + 0
Express 0 as 2abcd - 2abcd
(a^2 + b^2)(c^2 + d^2) = a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2 + 2abcd - 2abcd
Rewrite as:
(a^2 + b^2)(c^2 + d^2) = a^2c^2 + b^2d^2 - 2abcd + a^2d^2 + b^2c^2 + 2abcd
Group the terms
(a^2 + b^2)(c^2 + d^2) = (a^2c^2 + b^2d^2 - 2abcd) + (a^2d^2 + b^2c^2 + 2abcd)
Factorize each group
(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2
Hence, the two squares identity is (c) (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2
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