haya.Learning made clear
Rational expressions sheet
5-mark answer frame

Simplify 1/(x² − 5x + 6) + 1/(x² − 3x + 2).

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Write what is given and what is to be found. Given 1/(x² − 5x + 6) + 1/(x² − 3x + 2). To find: the sum in lowest form.
  2. 2 · Factorise each denominator.
    • x² − 5x + 6 = (x − 2)(x − 3)
    • x² − 3x + 2 = (x − 1)(x − 2)
  3. 3 · Write the LCM. LCM of the denominators = (x − 1)(x − 2)(x − 3).
  4. 4 · Rewrite and add. [(x − 1) + (x − 3)]/[(x − 1)(x − 2)(x − 3)] = (2x − 4)/[(x − 1)(x − 2)(x − 3)].
  5. 5 · Cancel common factors. 2x − 4 = 2(x − 2). Cancel (x − 2) to get 2/[(x − 1)(x − 3)].
  6. 6 · Check and box the answer. Try x = 4: 1/2 + 1/6 = 2/3, and 2/(3 × 1) = 2/3. Box: 2/[(x − 1)(x − 3)].

Scoring tip: Factorise every denominator before you find the LCM. A quick check with a number such as x = 4 catches sign mistakes.