Factorising quadratics

What factorising is

• Factorising is expanding in reverse — writing an expression as a product (things multiplied), usually two brackets.
• It is the key skill for solving quadratic equations and simplifying algebraic fractions later.
• A quadratic $ax^2 + bx + c$ is monic when $a = 1$ (just $x^2$) and non-monic when $a \neq 1$.
• Always look for a common factor first, before any other method.

Taking out a common factor

• A common factor is shared by every term — the highest number and any letters they all contain.
• Divide each term by it and write it outside a bracket:

$6x^2 + 9x = 3x(2x + 3)$

• Check by expanding: $3x(2x + 3) = 6x^2 + 9x$.

Factorising $x^2 + bx + c$ (monic)

• Find two numbers that multiply to $c$ and add to $b$, then put them in two brackets.
• The signs narrow the search:
  • $c$ positive, $b$ positive → both numbers positive.
  • $c$ positive, $b$ negative → both numbers negative.
  • $c$ negative → one positive and one negative.
• Example: $x^2 + 2x - 15 = (x + 5)(x - 3)$, since $5 \times (-3) = -15$ and $5 + (-3) = 2$.

Factorising $ax^2 + bx + c$ (non-monic)

• Use the splitting the middle term (grouping) method:
  • Multiply $a \times c$.
  • Find two numbers that multiply to $ac$ and add to $b$.
  • Split the middle $bx$ term into those two numbers.
  • Factorise in pairs (grouping) — both pairs must leave the same bracket.

Difference of two squares

• A difference of two squares (a square minus a square, no middle term) always factorises:

$a^2 - b^2 = (a - b)(a + b)$

• Both parts must be perfect squares. Example: $9x^2 - 25 = (3x)^2 - 5^2 = (3x - 5)(3x + 5)$.