How To Solve Quadratic Inequalities at GCSE Maths Level

June 9, 2021
Solving quadrantic inequalities

In September 2015, the GCSE Maths curriculum was updated to include new topics, including vectors, iterative methods and how to solve quadratic inequalities.

The British government wanted to bring the UK Maths GCSE in line with international standards and the demands of a changing job market. This motivated them to introduce new concepts and focus more on developing reasoning skills rather than just calculation

 

Let’s take a look at the expectations of the new GCSE maths curriculum by exploring a recently-introduced topic that pupils often struggle with: quadratic inequalities.

 

What are quadratic inequalities? 

Quadratic equations describe parabolic motion: a symmetrical plane curve that can be drawn in the shape of a U. Parabola often feature in real world problems in economics, physics and engineering.

A quadratic inequality is a second-degree equation that uses an inequality sign instead of an equal sign.

 

Solving a GCSE Maths quadratic inequality question 

In this article I am solving question nineteen of the June 2017 paper 3 (higher tier). The exam board is Pearson Edexcel.

This is not an easy task. It requires an understanding of the quadratic formula, as well as an understanding of substitution and the ability to sketch graphs. Unfortunately, there are no two ways about it: pupils dislike sketching graphs.

The question itself is simple and brief.

 

Solve 2x2 + 3x – 2 > 0

Here, I will explain the solution to this quadratic inequality in a few logical steps.

1) Firstly, we need to solve the quadratic equation by using the quadratic formula. We could try to factorise or use other methods, but it is better to avoid these techniques during exams.

Our aim is to sketch the graph of a parabola, which is a curve with determined properties, to obtain a mathematical solution from our plot.

2) quadrantic formula   is the quadrantic formula

 

3) At this point we need to remember that a quadratic equation has the form yax2 + bx +

In our case, a = 2,  b = 3,  c−2. By substituting into the quadratic formula, we obtain:

substituting into the quadrantic formula

4) By solving two equations we obtain the two points where the graph crosses the horizontal axis ( x axis). They are called roots.

quadrantic formula root 2

5) Things get a bit harder now as we need to remember that the orientation of the parabola is given by the sign of the a term.

If a > 0 the parabola is  ∪  shaped. If a < 0 the parabola is  ∩  shaped. In our case the sign of a is positive ( a = 2 ) thus our curve is  ∪  shaped.

 

6) Now things become even trickier as we need to sketch the graph. Here I am using a computer program, but I will lay out the underlying thinking as I go along.

The first thing we need to do is to sketch the axis and define on the horizontal axis ( x axis) the position of the points x1 and x2.

Thereafter, given that we know that the curve will be ∪ shaped, we can sketch the graph by connecting the points x1 and x2 and extending our curve toward infinity.

Then we need to shade the areas between the curve and the horizontal axis to visualise the solution.

inequality plot

We haven’t finished yet. We still need to write down the solution in mathematical terms, otherwise we will lose a mark. Looking at the shaded areas we can see that our parabola is greater than zero (the graph is above the horizontal axis) for the following values:

quadrantic formula solution

 

This question was worth three marks.

I hope that this article helps you master the tricky business of solving quadratic inequalities so that you can take on your Maths GCSE with confidence.

Federico Antonelli

Federico Antonelli

Maths tutor at TuitionWorks

I am an applied mathematician and qualified secondary teacher. I have done research in the field of nuclear energy and am currently studying toward a PhD with the University of Cranfield in aerospace materials.

Book Federico today →

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