O Level MathematicsE2.2 Algebraic manipulation (factorising quadratics, algebraic fractions, change of subject).

🔢 Crack the Code of Algebraic Manipulation!

Edudent Academy
12 Dec 25

Algebra is the language of patterns and logical thinking. Mastering it lets you **simplify complex expressions**, solve equations with ease, and secure those crucial marks in the O Level Extended paper. In this post we’ll zoom into factorising quadratics, tidying up algebraic fractions, and changing the subject—three super-powers every student needs.

Worked Example: Factorising & Rearranging

Problem: Given
2x2+5x−3x2−9=k\displaystyle \frac{2x^2+5x-3}{x^2-9} = k
, (i) factorise the numerator and denominator, (ii) simplify the fraction, and (iii) make
xx
the subject in terms of
kk
.

  • Step 1: Factorise the numerator:
    2x2+5x−3=(2x−1)(x+3)2x^2 + 5x - 3 = (2x - 1)(x + 3)
    .
  • Step 2: Factorise the denominator:
    x2−9=(x−3)(x+3)x^2 - 9 = (x - 3)(x + 3)
    .
  • Step 3: Cancel common factor
    (x+3)(x + 3)
    to get
    2x−1x−3=k\displaystyle \frac{2x - 1}{x - 3} = k
    .
  • Step 4: Change the subject. Cross-multiply:
    2x−1=k(x−3)2x - 1 = k(x - 3)
    .
  • Step 5: Expand and collect like terms:
    2x−1=kx−3k  ⇒  2x−kx=−3k+12x - 1 = kx - 3k \;\Rightarrow\; 2x - kx = -3k + 1
    .
  • Step 6: Factor out
    xx
    and divide:
    x(2−k)=1−3k  ⇒  x=1−3k2−kx(2 - k) = 1 - 3k \;\Rightarrow\; \boxed{\displaystyle x = \frac{1 - 3k}{2 - k}}
    .

Notice how careful factorisation simplifies the fraction before any rearrangement—saving time and minimising errors. **Practise** these steps with different quadratics and parameter values so you can glide through exam questions with confidence!