What is |x+y|^2+|x-y|^2 ?

The solution to |x+y|^2+|x-y|^2 is 2x^2+2y^2

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simplify |x+y|^2+|x-y|^2

simplify

∣ x + y ∣^{2} + ∣ x - y ∣^{2}

2 x^{2} + 2 y^{2}

Solution steps

∣ x + y ∣^{2} + ∣ x - y ∣^{2}

∣ x + y ∣^{2} = (x + y)^{2}

∣ x - y ∣^{2} = (x - y)^{2}

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

Expand

(x + y)^{2} : x^{2} + 2 x y + y^{2}

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

Expand

(x - y)^{2} : x^{2} - 2 x y + y^{2}

= x^{2} + 2 x y + y^{2} + x^{2} - 2 x y + y^{2}

Group like terms

= x^{2} + x^{2} + 2 x y - 2 x y + y^{2} + y^{2}

Add similar elements:

x^{2} + x^{2} = 2 x^{2}

= 2 x^{2} + 2 x y - 2 x y + y^{2} + y^{2}

Add similar elements:

2 x y - 2 x y = 0

= 2 x^{2} + y^{2} + y^{2}

Add similar elements:

y^{2} + y^{2} = 2 y^{2}

= 2 x^{2} + 2 y^{2}

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