Factorising Quadratic Expressions
Factorising quadratics is like being an algebraic detective – you’re working backwards to find the hidden pieces that were multiplied together. It’s a fundamental skill for simplifying complex problems and solving equations, used in everything from designing bridges to programming games.
The Two Key Methods
1. Factorising $x^2 + bx + c$
This is the most common type of quadratic. The goal is to find two numbers that:
- Multiply to make the constant term ($c$).
- Add to make the coefficient of $x$ ($b$).
2. Difference of Two Squares
This is a special case that’s quick to solve if you can spot it! Look for two perfect square terms separated by a minus sign.
$(a-b)(a+b)$
For example, $x^2 – 25$ becomes $(x-5)(x+5)$.
Worked Examples
Example 1: All Positives
Factorise $x^2 + 7x + 10$.
- We need two numbers that multiply to 10 and add to 7.
- Factor pairs of 10: (1, 10), (2, 5).
- Check which pair adds to 7: $2 + 5 = 7$. Perfect!
Answer: $(x+2)(x+5)$
Example 2: With Negatives
Factorise $x^2 – 8x + 12$.
- Multiply to +12, Add to -8. Since they multiply to a positive but add to a negative, both numbers must be negative.
- Factor pairs of 12: (-1, -12), (-2, -6), (-3, -4).
- Check which pair adds to -8: $-2 + (-6) = -8$.
Answer: $(x-2)(x-6)$
Tutor Insights
🤔 Common Misunderstandings
- Confusing signs: If the constant term ($c$) is positive, both numbers have the same sign. If $c$ is negative, they have different signs.
- Mixing up “multiply” and “add”: Remember: multiply to the end number, add to the middle number.
📝 Common Exam Mistakes
- Incorrect factor pairs: Choosing two numbers that don’t add up correctly. A systematic list helps avoid this.
- Sign errors: This is the most common error, especially when dealing with negative numbers.
- Forgetting to check: Always expand your answer at the end to see if it matches the original question.
Practice Questions
Factorise the following expressions.
- $x^2 + 9x + 20$
- $x^2 – 10x + 21$
- $x^2 + 5x – 14$
- $x^2 – 81$
Show Answers
- Answer: $(x+4)(x+5)$
- Answer: $(x-3)(x-7)$
- Answer: $(x-2)(x+7)$
- Answer: $(x-9)(x+9)$ (Difference of two squares)
FAQs
Why do we factorise?
A: Factorising is a key skill for simplifying expressions and is the first step in solving most quadratic equations (by setting each factor to zero). It’s a fundamental building block for more advanced algebra.
What’s the difference between factorising and solving?
A: Factorising turns an expression like $x^2 + 7x + 10$ into a product of factors, $(x+2)(x+5)$. Solving takes an equation like $x^2 + 7x + 10 = 0$ and finds the values of $x$ that make it true (in this case, $x=-2$ and $x=-5$).
