**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.

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 $$ \frac{-b}{2a} - i \frac{\sqrt{-\Delta}}{2a} $$

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?

Show Solution

**Example 2**

**Question**

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

Show Solution

**More Advanced Questions**

**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?

Show Solution

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 ().

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