What is 1/(x-2y)-(x^2+4y^2)/(x^3-8y^3) ?

The solution to 1/(x-2y)-(x^2+4y^2)/(x^3-8y^3) is (2xy)/(x^3-8y^3)

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simplify 1/(x-2y)-(x^2+4y^2)/(x^3-8y^3)

simplify

\frac{1}{x - 2 y} - \frac{x ^{2} + 4 y ^{2}}{x ^{3} - 8 y ^{3}}

\frac{2 x y}{x ^{3} - 8 y ^{3}}

Solution steps

\frac{1}{x - 2 y} - \frac{x ^{2} + 4 y ^{2}}{x ^{3} - 8 y ^{3}}

Least Common Multiplier of

x - 2 y, x^{3} - 8 y^{3} : (x - 2 y) (x^{3} - 8 y^{3})

Adjust Fractions based on the LCM

= \frac{x ^{3} - 8 y ^{3}}{( x - 2 y ) ( x ^{3} - 8 y ^{3} )} - \frac{( x ^{2} + 4 y ^{2} ) ( x - 2 y )}{( x - 2 y ) ( x ^{3} - 8 y ^{3} )}

Apply the fraction rule:

\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}

= \frac{x ^{3} - 8 y ^{3} - ( x ^{2} + 4 y ^{2} ) ( x - 2 y )}{( x - 2 y ) ( x ^{3} - 8 y ^{3} )}

Simplify

x^{3} - 8 y^{3} - (x^{2} + 4 y^{2}) (x - 2 y) : 2 x^{2} y - 4 x y^{2}

= \frac{2 x ^{2} y - 4 x y ^{2}}{( x - 2 y ) ( x ^{3} - 8 y ^{3} )}

Cancel

\frac{2 x ^{2} y - 4 x y ^{2}}{( x - 2 y ) ( x ^{3} - 8 y ^{3} )} : \frac{2 x y}{x ^{3} - 8 y ^{3}}

= \frac{2 x y}{x ^{3} - 8 y ^{3}}

1 < x < 4

\frac{2 x}{9} = - 65

36 x^{2} - 14 x - 2

5 y^{2} - 35 = 0

\frac{x}{6} = \frac{3}{9}

What is 1/(x-2y)-(x^2+4y^2)/(x^3-8y^3) ?
The solution to 1/(x-2y)-(x^2+4y^2)/(x^3-8y^3) is (2xy)/(x^3-8y^3)