(b) Hence, or otherwise, solve 6x² – 5x – 4 < 0
Quadratic expressions and inequalities are a core part of the GCSE Maths curriculum. In this post, we'll walk through a typical two-part exam question, breaking down each step clearly. Mastering these skills is essential for your exams with AQA, Edexcel, or OCR.
The Exam Question
We are going to solve the following:
> (a) Factorise 6x² - 5x - 4
> (b) Hence, or otherwise, solve 6x² - 5x - 4 < 0
The word "hence" is a big clue! It tells us that our answer from part (a) will make solving part (b) much quicker and easier.
Part (a): Factorising 6x² - 5x - 4
Factorising a quadratic in the form ax² + bx + c where a ≠ 1 (here, a = 6) requires a methodical approach. We are looking for two binomials: (px + q)(rx + s).
Step 1: Multiply a and c.
Here, a = 6 and c = -4.
a × c = 6 × (-4) = -24.
Step 2: Find two numbers that multiply to -24 and add to b (-5).
We need m and n such that m × n = -24 and m + n = -5.
Let's list factor pairs of 24: (1,24), (2,12), (3,8), (4,6).
Since the product is negative (-24), one number must be positive and one negative. To get a sum of -5, we need the larger number (in absolute value) to be negative.
The pair 3 and -8 works: 3 × (-8) = -24 and 3 + (-8) = -5. Perfect!
Step 3: Rewrite the middle term using these numbers.
We split the -5x into 3x and -8x.
6x² - 5x - 4 = 6x² + 3x - 8x - 4
Step 4: Factorise in pairs.
Group the first two terms and the last two terms, and factor out the common factor from each.
(6x² + 3x) + (-8x - 4)
From the first group: 3x(2x + 1).
From the second group: -4(2x + 1).
Notice (2x + 1) is a common factor!
- Step 5: Factor out the common binomial.
3x(2x + - 1) - 4(2x +
- 1) = (2x + 1)(3x - 4)
✅ Answer for (a): (2x + 1)(3x - 4)
Always check by expanding: (2x+1)(3x-4) = 6x² - 8x + 3x - 4 = 6x² - 5x - 4. Correct!
Part (b): Solving 6x² - 5x - 4 < 0
Now we use our factorisation. The inequality becomes:
(2x + 1)(3x - 4) < 0
This product is less than zero, meaning it is negative. For the product of two things to be negative, one must be positive and the other must be negative.
Step 1: Find the critical values.
These are the values of x where the expression equals zero. Set each factor to zero:
2x + 1 = 0 \quad \Rightarrow \quad x = -¹⁄2
3x - 4 = 0 \quad \Rightarrow \quad x = ⁴⁄3
These two values split the number line into three intervals.
Step 2: Test the sign of the product in each interval.
We can use a quick sign chart or sketch. The critical values are x = -0.5 and x ≈ 1.333.
The intervals are:
x < -¹⁄2
-¹⁄2 < x < ⁴⁄3
- x > ⁴⁄3
Pick a test value from each interval and substitute it into the factorised form (2x+1)(3x-4). We only care if the result is positive or negative.
- Interval 1: Let x = -1.
2(-1)+1 = -1 (negative)
3(-1)-4 = -7 (negative)
Negative × Negative = Positive. ❌ Not less than 0.
- Interval 2: Let x = 0.
2(0)+1 = 1 (positive)
3(0)-4 = -4 (negative)
Positive × Negative = Negative. ✅ This satisfies < 0.
- Interval 3: Let x = 2.
2(2)+1 = 5 (positive)
3(2)-4 = 2 (positive)
Positive × Positive = Positive. ❌ Not less than 0.
Step 3: Consider the endpoints.
The inequality is strictly less than (<), not ≤. This means we do not include the points where the expression equals zero. So, x = -¹⁄2 and x = ⁴⁄3 are not included in the solution.
✅ Answer for (b): -¹⁄2 < x < ⁴⁄3
This means any x between -0.5 and 1.333... (but not including those numbers) will make the original quadratic expression negative.
flowchart TD
A["Start: Solve 6x² - 5x - 4 < 0"] --> B["Factorise: (2x+1)(3x-4)"];
B --> C["Find Critical Values: x = -1/2, x = 4/3"];
C --> D["Plot on Number Line"];
D --> E["Test Interval x < -1/2: Product +ve"];
D --> F["Test Interval -1/2 < x < 4/3: Product -ve"];
D --> G["Test Interval x > 4/3: Product +ve"];
F --> H["Solution: -1/2 < x < 4/3"];
Key Takeaways for Your GCSE Exam
Master the Factorising Method: The ac method (multiply a and c, find factors that add to b) is reliable for harder quadratics.
"Hence" is a Command: It almost always means you must use your previous answer to save time and gain full marks.
Solving Quadratic Inequalities:
Factorise the expression.
Find the critical values (roots).
Use a number line or sign chart to test intervals.
- Pay close attention to whether the inequality is strict (<, >) or inclusive (≤, ≥) when writing your final answer.
- Practise this process with different quadratics to build your confidence. Good luck with your revision!