Why factorising quadratics matters
Ever wondered why some math problems feel like puzzles you can crack with a simple trick? Factorising turns a tough equation into a friendly product you can split apart.
💡 In Simple Words: A quadratic equation is an expression where the highest power of x is 2, like ax²+bx+c=0. By factorising we rewrite it as a product of two simpler expressions, then use the zero‑product rule (if AB=0, then A=0 or B=0) to find x.
What is a quadratic equation?
A quadratic equation looks like ax² + bx + c = 0 where a, b, c are numbers and a ≠ 0. Think of it as a hill‑shaped curve on a graph; the points where it crosses the x‑axis are the solutions.
How to solve by factorisation – the 5‑step recipe
- Step 1: Bring everything to one side so the equation equals 0.
- Step 2: Look for two numbers that multiply to a·c (product of a and c) and add to b.
- Step 3: Split the middle term bx using those two numbers and factor by grouping.
- Step 4: Apply the zero‑product rule – set each bracket equal to 0.
- Step 5: Solve the resulting simple linear equations.
Worked example 1
Solve x² – 5x + 6 = 0.
Step 1: It’s already =0.
Step 2: We need two numbers that multiply to 6 (a·c) and add to –5. Those are –2 and –3.
Step 3: Rewrite –5x as –2x –3x:
x² – 2x – 3x + 6 = 0
Step 4: Group and factor:
(x² – 2x) + (‑3x + 6) = 0
x(x – 2) –3(x – 2) = 0
Now we have a common factor (x – 2):
(x – 3)(x – 2) = 0
Step 5: Set each bracket to 0:
x – 3 = 0 → x = 3
x – 2 = 0 → x = 2
So the solutions are x = 2 and x = 3.
Worked example 2 (with a leading coefficient ≠ 1)
Solve 2x² + 7x + 3 = 0.
Step 1: Already =0.
Step 2: a·c = 2×3 = 6. We need two numbers that multiply to 6 and add to 7 – they are 1 and 6.
Step 3: Split 7x:
2x² + 1x + 6x + 3 = 0
Step 4: Group:
(2x² + x) + (6x + 3) = 0
x(2x + 1) + 3(2x + 1) = 0
Factor out the common binomial (2x + 1):
(x + 3)(2x + 1) = 0
Step 5: Solve:
x + 3 = 0 → x = –3
2x + 1 = 0 → x = –½
Solutions: x = –3 and x = –½.
Quick reference table
| Step | What to do | Why it works |
|---|---|---|
| 1 | Move all terms to one side → 0 | Standard form for factorisation |
| 2 | Find two numbers with product a·c and sum b | These numbers split the middle term |
| 3 | Rewrite and group | Creates a common factor |
| 4 | Factor out the common binomial | Turns the quadratic into a product |
| 5 | Apply zero‑product rule | Each factor set to zero gives a solution |
Common pitfalls to avoid
- Forgetting to make the constant term zero before factorising.
- Mixing up the signs when the product a·c is negative.
- Skipping the grouping step – it’s the bridge to factorisation.
📝 Likely Exam Questions
- Solve x² – 4x – 5 = 0.
Answer: (x – 5)(x + 1)=0 → x=5 or x=‑1. - Factorise and solve 3x² – 11x + 6 = 0.
Answer: (3x – 2)(x – 3)=0 → x=2/3 or x=3. - Given that (x – 2) is a factor of 2x² – kx + 8, find k.
Answer: Substitute x=2 → 2·4 – 2k + 8 =0 → 8 – 2k + 8 =0 → k=8. - Write the quadratic equation whose roots are –1 and 4.
Answer: (x +1)(x –4)=0 → x² –3x –4 =0. - Solve 4x² – 12x + 9 = 0 by factorisation.
Answer: (2x –3)²=0 → x = 3/2 (double root).