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Complex Plane

Simplifyx2+y2...

Question

Simplify
$(x^{2} + y^{2} - \frac{2 x y}{5}) - (- \frac{4 x^{2}}{3} - \frac{4 y^{2}}{5} + \frac{8 x y}{4})$

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Solution

Consider $= (x^{2} + y^{2} - \frac{2 x y}{5}) - (- \frac{4 x^{2}}{3} - \frac{4 y^{2}}{5} + \frac{8 x y}{4})$
$= x^{2} + y^{2} + \frac{2 x y}{5} + \frac{4 x^{2}}{3} + \frac{4 y^{2}}{5} - \frac{8 x y}{4}$
$= (x^{2} + \frac{4 x^{2}}{3}) + (y^{2} + \frac{4 y^{2}}{5}) + (\frac{2 x y}{5} - \frac{8 x y}{4})$
Taking LCM
$= (\frac{3 x^{2} + 4 x^{2}}{3}) + (\frac{5 y^{2} + 4 y^{2}}{5}) + (\frac{8 x y - 40 x y}{20}) = (\frac{7 x^{2}}{3}) + (\frac{9 y^{2}}{5}) + (\frac{- 32 x y}{20}) = \frac{7 x^{2}}{3} + \frac{9 y^{2}}{5} - \frac{32 x y}{20}$
Hence, $(x^{2} + y^{2} - \frac{2 x y}{5}) - (- \frac{4 x^{2}}{3} - \frac{4 y^{2}}{5} + \frac{8 x y}{4})$ is equal to $\frac{7 x^{2}}{3} + \frac{9 y^{2}}{5} - \frac{32 x y}{20}$

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