Learning Goal

Syllabus Requirement

Students need to learn the use and manipulation of surds, with the below point being highlighted in particular:

•    Be able to rationalise denominators.

Study Notes

Recall that the quadratic equation for ??ax^{2} + bx + c = 0?? is: $$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$ The discriminant is the name given to the term ??b^{2} - 4ac??, in the quadratic equation. The discriminant is often referred to as ??D?? or ??\Delta??.

The value of the discriminant reveals information about the roots of the quadratic equation and the three cases are listed in the information box below, note that from now on we will use the symbol ??\Delta?? to refer to the discriminant.

The 3 Cases for the Discriminant

Case 1: Positive Discriminant
If the discriminant is more than 0, then there are 2 distinct, real roots: $$\frac{-b + \sqrt{\Delta}}{2a}$$ and $$\frac{-b - \sqrt{\Delta}}{2a}$$ Case 2: Zero Discriminant
If the discriminant is 0, then there is 1 real root: $$\frac{-b}{2a}$$ Case 3: Negative Discriminant
If the discriminant is less than 0, then there are no real roots. Note that there are 2 complex roots in this case, which are shown below: $$\frac{-b}{2a} + i \frac{\sqrt{-\Delta}}{2a}$$ and $$\frac{-b}{2a} - i \frac{\sqrt{-\Delta}}{2a}$$

Worked Examples

We'll now run through some examples of the kinds of questions that you may see on the exam:

Finding the Number of Roots

Example 1

Question

How many real roots does the following quadratic equation, ??x^{2} - 3x + 4 = 0??, have?

Example 2

Question

How many real roots does the following quadratic equation, ??-3x^{2} + x - 7 = 0??, have?

Example 3

Question

For, ??2x^{2} - kx + 6 = 0??, find the ranges of values of ??k?? for which there are 2 real roots, 1 real root and no real roots?

Exam Questions

The table below contains every exam question that has been asked on this topic, this includes normal papers from both January and June sittings, International papers and Specimen papers.

To see an exam question and solution simply click on the load question icon (), the question will appear below the table and the solution can be shown by clicking the "Show Solution" button that also appears.

The full exam paper and mark schemes are also available for download by clicking on the download icons in each row of the table ().

Exam Board Subject Paper Year Month Module Question
No.
Parts Total
Marks
Question
Exam
Paper
Mark
Scheme
Edexcel Maths Standard 2005 January Core 1 3 n/a 4
Edexcel Maths Standard 2006 January Core 1 10 (c), (d) 6
Edexcel Maths Standard 2006 June Core 1 8 (a), (b) 6
Edexcel Maths Standard 2007 January Core 1 5 n/a 4
Edexcel Maths Standard 2008 January Core 1 8 (a), (b) 7
Edexcel Maths Standard 2008 June Core 1 8 (a), (b) 5
Edexcel Maths Standard 2009 January Core 1 7 (a), (b) 7
Edexcel Maths Standard 2009 June Core 1 6 n/a 4
Edexcel Maths Standard 2010 January Core 1 10 (b) 4
Edexcel Maths Standard 2010 June Core 1 4 (c) 2
Edexcel Maths Standard 2011 January Core 1 8 (a), (b) 7
Edexcel Maths Standard 2011 June Core 1 7 (a), (b), (c) 6
Edexcel Maths Standard 2012 June Core 1 8 (a), (b) 5
Edexcel Maths Standard 2013 January Core 1 9 (a), (b) 7
Edexcel Maths Standard 2013 June Core 1 10 (b) 3

Exam Tips

1. You should know the three possible cases for the discriminant as these will nearly always be examined.

2. Often you will be required to find the range of possible values of some constant, that will give a discriminant of the required form. You should be comfortable in solving inequalities, including quadratic inequalities that appear on some harder discriminant questions.