Factorisation of Quadratics: Explained with Examples

A quadratic expression has the form ax squared + bx + c, where a is not zero. Factorisation of quadratics means rewriting this expression as a product of two linear factors, like (px + q)(rx + s). Once a quadratic is factorised, its roots can be read off almost instantly, which is why this skill sits at the heart of algebra.

This matters because quadratic relationships appear everywhere: the path of a thrown ball, the area of a rectangle, profit curves in business. Before solving these real problems, you must break the quadratic into its factors quickly and reliably.

Key Concepts

• Quadratic: a polynomial of degree 2, written in standard form ax squared + bx + c with a not equal to zero.
• Monic quadratic: a quadratic where the leading coefficient a equals 1, for example x squared + 7x + 12.
• Product-sum rule: for a monic quadratic x squared + bx + c, find two numbers whose product is c and whose sum is b.
• Splitting the middle term: for a non-monic quadratic, rewrite bx as two terms whose coefficients multiply to a times c and add to b.

How It Works

1. If the quadratic is monic, find two numbers with product c and sum b, then write (x + m)(x + n).
2. If a is not 1, multiply a by c and find two numbers with that product and sum b.
3. Use those two numbers to split the middle term into two terms.
4. Now factorise by grouping: take out the GCF of each pair and factor out the common bracket.
5. Expand your answer to verify it gives back the original quadratic.

Worked Example

Factorise 6x squared + 7x + 2.

1. Here a = 6 and c = 2, so a times c = 12. Find two numbers with product 12 and sum 7: they are 3 and 4.
2. Split the middle term: 6x squared + 3x + 4x + 2.
3. Group: (6x squared + 3x) + (4x + 2) = 3x(2x + 1) + 2(2x + 1).
4. Factor out the common bracket: (3x + 2)(2x + 1).

Check by expanding: (3x + 2)(2x + 1) = 6x squared + 3x + 4x + 2 = 6x squared + 7x + 2. It matches, so the factorisation is correct.

Common Misconceptions

• The two numbers must be factors of b. Wrong. Their product must equal a times c (or just c for a monic quadratic) and their sum must equal b. For 6x squared + 7x + 2, the numbers 3 and 4 are not factors of 7.
• Signs can be guessed at the end. Wrong. Signs matter from the start: for x squared - 5x + 6 you need two negative numbers, giving (x - 2)(x - 3), because (-2) times (-3) = 6 and (-2) + (-3) = -5.
• Every quadratic factorises with integers. Wrong. Some quadratics, like x squared + x + 1, have no integer factors at all and need the quadratic formula instead.

Real-World Applications

• In physics, the height of a projectile is a quadratic in time; factorising it tells you exactly when the object hits the ground.
• In geometry, area problems such as finding the sides of a rectangle from its area and perimeter lead to quadratics that must be factorised.
• In business, profit and revenue are often modelled as quadratics, and their factors reveal the break-even points.

To recap: monic quadratics use the product-sum rule; others need the middle term split with product a times c and sum b, then grouping. Expand your factors to confirm.