Sum of algebraic fractions

Statement

Write as a single fraction: (a) $\dfrac{1}{y^2} - \dfrac{1}{y^2 + 2}$; (b) $\dfrac{4x}{x(x - 2)} + \dfrac{3}{x}$.

Solution

(a) The least common multiple is $y^2(y^2 + 2)$: $\frac{1}{y^2} - \frac{1}{y^2 + 2} = \frac{(y^2 + 2) - y^2}{y^2(y^2 + 2)} = \frac{2}{y^2(y^2 + 2)}.$

(b) First simplify $\dfrac{4x}{x(x - 2)} = \dfrac{4}{x - 2}$ (with $x \neq 0$). With common denominator $x(x - 2)$: $\frac{4}{x - 2} + \frac{3}{x} = \frac{4x + 3(x - 2)}{x(x - 2)} = \frac{7x - 6}{x(x - 2)}.$

$\boxed{\dfrac{2}{y^2(y^2 + 2)}\ ;\quad \dfrac{7x - 6}{x(x - 2)}}$