Factorising Quadratics

Factorising Simple Quadratics

Quadratic expressions come in the form ax^2 + bx + c. Factorising such an expression means writing it as two brackets that multiply together to give the original expression.
Identify ‘a’, ‘b’ and ‘c’ in the quadratic equation. The coefficient of x^2 is ‘a’, the coefficient of x is ‘b’, and ‘c’ is the constant.
To factorise, find two numbers that multiply to give ‘ac’ (a times c), and add to give ‘b’.

A special case of factorising involves expressions of the form a^2 - b^2, which is known as the difference of two squares.
The expression a^2-b^2 can be factorised as (a+b)(a-b). For instance, if given x^2 - 9, it can be factorised as (x+3)(x-3).

When ‘a’ is not equal to 1, the quadratic is said to be a complex quadratic. Factorising complex quadratics involves the same principle, but usually requires an additional factorising step after identifying the two numbers that satisfy the conditions.
One way of factorising complex quadratics is by using the method of cross-multiplication. Although complex, it is effective and it can handle any type of quadratic expression, even those where ‘a’ ≠ 1.

After correctly factorising a quadratic expression, expanding the brackets should return the original expression. This is a useful means of checking your work.
Factorising quadratics is an essential tool for solving quadratic equations and inequalities, and for sketching parabola graphs.
Remember not every quadratic can be factorised into neat whole numbers. If you can’t find factors which suitably match the rules specified, then analytical methods such as the quadratic formula or numerical methods may be needed to locate the roots of the equation.