Algebraic Fractions

Simplifying

Factorise numerator and denominator, then cancel common factors.

$\frac{x^2 - 4}{x + 2} = \frac{(x+2)(x-2)}{x+2} = x - 2$

Adding/Subtracting

Find a common denominator.

$\frac{3}{x+1} + \frac{2}{x-1} = \frac{3(x-1) + 2(x+1)}{(x+1)(x-1)} = \frac{5x - 1}{x^2 - 1}$

Multiplying

Multiply numerators and denominators. Simplify first if possible.

$\frac{x}{x+3} \times \frac{x+3}{x^2} = \frac{1}{x}$

Dividing

Flip the second fraction and multiply.

$\frac{x^2}{4} \div \frac{x}{2} = \frac{x^2}{4} \times \frac{2}{x} = \frac{x}{2}$

Solving Equations with Algebraic Fractions

Multiply through by the common denominator to clear fractions.

$\frac{3}{x} + \frac{1}{2} = 2$ → Multiply by $2x$: $6 + x = 4x$ → $x = 2$.