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If x = \sqrt{2006} and y = \sqrt{2007}, what is the simplified value of (x + y)^2 + (x - y)^2?

Jun 11, 2021

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If

$$x = \sqrt{2006}$$ and $$y = \sqrt{2007}$$,
what is the simplified value of

$$(x + y)^2 + (x - y)^2$$?

$$\begin{array}{|rcll|} \hline (x + y)^2 + (x - y)^2 &=& x^2+2xy+y^2+x^2-2xy+y^2 \\ &=& x^2+y^2+x^2+y^2 \\ &=& 2x^2+2y^2 \\ &=& 2(x^2+y^2) \quad | \quad x^2=2006,~y^2=2007 \\ &=& 2(2006+2007) \\ &=& 2*2013 \\ \mathbf{ (x + y)^2 + (x - y)^2 } &=& \mathbf{4026} \\ \hline \end{array}$$

Jun 11, 2021