Factorising Harder Quadratics
Factorising quadratics in the form $ax^2 + bx + c$ (where $a$ is not 1) is a key higher-tier skill. It’s used to solve complex equations and is essential for designing everything from bridges to rollercoasters. This guide will walk you through the most reliable method to master them.
The AC Method: Step-by-Step
The most systematic way to factorise harder quadratics is the AC method, also known as “splitting the middle term”. Let’s walk through it with the example: Factorise $2x^2 + 7x + 3$.
Step 1: Identify a, b, and c
In $2x^2 + 7x + 3$:
$a = 2$, $b = 7$, $c = 3$
Step 2: Calculate ac
Multiply the coefficient of $x^2$ by the constant term.
$ac = 2 \times 3 = 6$
Step 3: Find Two Numbers
Find two numbers that multiply to $ac$ (6) and add to $b$ (7).
The numbers are 1 and 6.
Step 4: Split the Middle Term
Rewrite the middle term ($7x$) using the two numbers found.
$2x^2 + \mathbf{1x + 6x} + 3$
Step 5: Factorise by Grouping
Group the first two terms and the last two, then factorise each pair.
$(2x^2 + x) + (6x + 3)$
$= x(2x + 1) + 3(2x + 1)$
Step 6: Write the Final Answer
The repeated bracket is one factor; the terms outside form the other.
$(2x + 1)(x + 3)$
Worked Examples
Example 1: Signs are Negative
Factorise $3x^2 – 10x + 8$.
- $a = 3$, $b = -10$, $c = 8$. So $ac = 24$.
- Find two numbers that multiply to 24 and add to −10. Both must be negative: −4 and −6.
- Split: $3x^2 – 4x – 6x + 8$.
- Group: $(3x^2 – 4x) + (-6x + 8)$.
- Factorise each pair: $x(3x – 4) – 2(3x – 4)$.
Answer: $(3x – 4)(x – 2)$
Example 2: Mixed Signs
Factorise $5x^2 + 13x – 6$.
- $a = 5$, $b = 13$, $c = -6$. So $ac = -30$.
- Find two numbers that multiply to −30 and add to 13. They must have different signs: 15 and −2.
- Split: $5x^2 + 15x – 2x – 6$.
- Group: $(5x^2 + 15x) + (-2x – 6)$.
- Factorise each pair: $5x(x + 3) – 2(x + 3)$.
Answer: $(x + 3)(5x – 2)$
Tutor Insights
🤔 Common Misunderstandings
- Forgetting ‘a’ in ‘ac’: The most common error is looking for factors of $c$ alone. You must multiply $a$ and $c$ first!
- Sign errors: Mixing up positive and negative factors. A systematic list of factor pairs helps prevent this.
- Incorrect grouping: Not taking out the correct common factor from the second pair, or not making the brackets match.
📝 Common Exam Mistakes
- Not simplifying first: If all terms share a common factor (e.g., $6x^2 + 9x – 6$), factor it out first to reduce the numbers.
- Arithmetic mistakes: Simple errors when calculating $ac$ or adding factors.
- Not checking the answer by expanding the brackets at the end.
Practice Questions
Factorise the following expressions.
- $2x^2 + 5x + 2$
- $3x^2 + 11x + 6$
- $4x^2 – 11x + 6$
- $6x^2 – 7x – 5$
Show Answers
- $(2x + 1)(x + 2)$
- $(3x + 2)(x + 3)$
- $(4x – 3)(x – 2)$
- $(3x – 5)(2x + 1)$
FAQs
Q: Can I use another method?
A: The AC method is the most reliable and systematic. Some people use trial and error, but this can be slow and frustrating with larger numbers. For solving equations, the quadratic formula is an alternative, but factorising is often quicker when it works.
Q: What if it doesn’t factorise?
A: Not all quadratics factorise into neat brackets with whole numbers. If you’ve tried the AC method and can’t find two numbers that work, it likely doesn’t factorise this way. In a GCSE exam question, you can usually assume it will factorise unless told otherwise.
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