Factorising Quadratic Expressions

Learn how to factorise quadratic expressions using splitting the middle term, perfect squares, special cases, examples, and student-friendly notes.

1. Introduction

A quadratic expression is an expression of the form:

\(ax^2 + bx + c\)

where \(a, b, c\) are numbers and \(a \neq 0\). The most common method of factorising quadratics is the middle-term splitting method.

2. Understanding the Logic

To factorise a quadratic, we try to write it as a product of two binomials:

\(ax^2 + bx + c = (px + q)(rx + s)\)

The key idea: find two numbers that:

  • Multiply to \(ac\)
  • Add to \(b\)

3. Steps to Factorise a Quadratic Expression

  1. Identify \(a, b, c\) from \(ax^2 + bx + c\).
  2. Compute the product \(ac\).
  3. Find two numbers whose product is \(ac\) and sum is \(b\).
  4. Split the middle term using these two numbers.
  5. Group the terms into pairs.
  6. Factorise each group separately.
  7. Take out the common binomial factor.

3.1. Example Walkthrough

Factorise: \(x^2 + 7x + 12\)

Step 1: \(a = 1, b = 7, c = 12\)

Step 2: Product = \(ac = 12\)

Step 3: Find two numbers that multiply to 12 and add to 7 → 3 and 4

Step 4: Split middle term:

\(x^2 + 3x + 4x + 12\)

Step 5: Group:

\((x^2 + 3x) + (4x + 12)\)

Step 6: Factorise each group:

  • \(x(x + 3)\)
  • \(4(x + 3)\)

Final Answer:

\((x + 3)(x + 4)\)

4. Special Cases

Some quadratics match special patterns:

4.1. 1. Perfect Square Trinomials

If the quadratic is of the form:

\(a^2 + 2ab + b^2 = (a + b)^2\)

or

\(a^2 - 2ab + b^2 = (a - b)^2\)

Example:

\(x^2 + 10x + 25 = (x + 5)^2\)

4.2. 2. When \(c = 0\)

If the expression is:

\(ax^2 + bx\)

Take out \(x\):

\(x(ax + b)\)

Example:

\(6x^2 - 9x = 3x(2x - 3)\)

4.3. 3. When \(a = 1\) (Simplest Case)

Quadratics like:

\(x^2 + bx + c\)

Look for two numbers multiplying to \(c\) and adding to \(b\).

Example:

\(x^2 + 8x + 15 = (x + 3)(x + 5)\)

5. More Examples

1. Factorise: \(2x^2 + 5x + 3\)

Product = \(6\), Sum = \(5\) → 2 and 3

Answer: \((2x + 3)(x + 1)\)

2. Factorise: \(x^2 - 9x + 20\)

Product = 20, Sum = -9 → -5 and -4

Answer: \((x - 5)(x - 4)\)

3. Factorise: \(3x^2 - 11x + 10\)

Product = 30, Sum = -11 → -5 and -6

Answer: \((3x - 5)(x - 2)\)

6. Common Mistakes

  • Choosing wrong pair of numbers for splitting the middle term.
  • Not grouping properly after splitting.
  • Forgetting to factorise fully (stopping halfway).
  • Mixing signs when splitting the middle term.
  • Not checking the final product.

7. Quick Practice

Factorise:

  1. \(x^2 + 6x + 8\)
  2. \(2x^2 - 7x + 3\)
  3. \(x^2 - 3x - 10\)
  4. \(4x^2 + 4x - 8\)

8. Summary

  • Quadratics are expressions of the form \(ax^2 + bx + c\).
  • The middle-term splitting method is the most common technique.
  • Find two numbers whose product is \(ac\) and sum is \(b\).
  • Special cases include perfect squares and cases where \(c = 0\).