Factorising Quadratics

Understanding Quadratic Equations

A quadratic equation is an equation that can be written in the form ax² + bx + c = 0.
The highest power of the variable (often denoted x) in the equation is 2, which makes it ‘quadratic’.
In ax² + bx + c = 0, a, b, and c are coefficients, and these can be any real number.

Factorising Quadratics

Factorising is the process of splitting the quadratic equation into two separate brackets (also known as binomial expressions).
When a quadratic equation is factorised, it takes the form (px + q)(rx + s) = 0.
p, q, r, and s are also coefficients that are determined during the factorisation process.

Quadratics with a Leading Coefficient of 1

If the coefficient of x² (a) is 1, then the quadratic equation can be factorised by finding two numbers that multiply to give c (the constant term) and add up to give b (the coefficient of x).
For example, the quadratic equation x² + 5x + 6 = 0 can be factorised into (x + 2)(x + 3) = 0. Here, 2 and 3 multiply to give 6 (c) and add to give 5 (b).

Quadratics with a Leading Coefficient Not Equal to 1

If a is not equal to 1, the quadratic equation takes the form ax² + bx + c = 0. Such quadratics are slightly more difficult to factorise.
To factorise, first multiply a and c. Then find two numbers that multiply to give the product of a and c and add up to give b.
These numbers replace the original bx term, then factor by grouping is used to complete the factorisation.
For example, the quadratic equation 2x² + 11x + 5 = 0 can be factorised into (2x + 1)(x + 5) = 0.

Solving Quadratic Equations

Once a quadratic equation has been factorised into the form (px + q)(rx + s) = 0, it can be solved by setting each factor equal to zero and then solving for x.
For instance, if (x + 2)(x + 3) = 0, then x + 2 = 0 or x + 3 = 0. Solving the equations gives the x-values of -2 and -3.

Checking Your Factorisation

Always check your factorisation by expanding the brackets. If you get back to the original quadratic, your factorisation is correct.
This is a key step in ensuring you have correctly factorised and solved the quadratic equation.

Factorising is a critical skill for solving quadratic equations and also plays a key role in higher mathematical studies, especially algebra. Regular practice of factorising will help make the process more familiar and automatic.