Exam questions are built to simplify, so an algebraic fraction given to you almost always has a common factor hiding in its numerator and denominator. If you can factorise the easier side first, it tells you what factor to look for on the harder side.

Worked example: Simplify $\dfrac{4x+6}{2x^2-7x-15}$.

Numerator factors easily: $2(2x+3)$. That strongly suggests $(2x+3)$ is a factor of the denominator too — dividing confirms $(2x^2-7x-15)=(2x+3)(x-5)$, so the fraction simplifies to $\dfrac{2}{x-5}$.

Factorise everything first, cancel common factors (vertically or diagonally across the two fractions), then multiply what's left. For division, use "keep, change, flip" — keep the first fraction, change ÷ to ×, flip the second fraction — then proceed as a multiplication.

Worked example: $\dfrac{x^2-9}{x+4}\times\dfrac{x+4}{3x+9}$. Factorise: $\dfrac{(x-3)(x+3)}{x+4}\times\dfrac{x+4}{3(x+3)}$. Cancel $(x+4)$ diagonally and $(x+3)$ vertically, leaving $\dfrac{x-3}{3}$.

Factorise every denominator before finding the common denominator — this often reveals that the "lowest" common denominator is much smaller than it first looks.

Worked example: $\dfrac{3}{x-1}-\dfrac{2}{x^2-1}$. Since $x^2-1=(x-1)(x+1)$, the LCD is just $(x-1)(x+1)$ — not the cubic you'd get by multiplying the denominators blindly.

Multiply every term by the LCD to clear all the fractions at once, then solve whatever's left — which might be linear or quadratic.

Linear case: Solve $3-\dfrac{5}{x+1}=1$.

Multiply by $(x+1)$: $3(x+1)-5=1(x+1) \Rightarrow 3x-2=x+1 \Rightarrow x=1.5$.

Quadratic case: Solve $\dfrac{9x+4}{x}=-2x$.

Multiply by $x$: $9x+4=-2x^2 \Rightarrow 2x^2+9x+4=0 \Rightarrow (2x+1)(x+4)=0 \Rightarrow x=-\tfrac12$ or $x=-4$.

Solve $\dfrac{4}{x-1}-\dfrac{3}{x+2}=1$.

1. Multiply every term by $(x-1)(x+2)$: $4(x+2)-3(x-1)=(x-1)(x+2)$.
2. Expand carefully: $4x+8-3x+3 = x^2+x-2$, giving $x+11=x^2+x-2$.
3. Simplify: subtract $x$ from both sides → $11=x^2-2$.
4. Rearrange to standard form: $x^2-13=0 \Rightarrow x^2=13$.
5. $x=\sqrt{13}$ or $x=-\sqrt{13}$.

• Cancelling individual terms instead of whole factors.
• Dropping the sign on a subtracted numerator when it isn't bracketed.
• Multiplying denominators together for the LCD without factorising first.
• Reaching an $x^2$ equation and not switching to quadratic-solving methods.