Discriminants
What is the discriminant of a quadratic function?
- The discriminant of a quadratic is often denoted by the Greek letter (upper case delta)
- For a quadratic the discriminant is given by
- The discriminant is the expression that is inside the square root in the quadratic formula
- This is not on the exam formula sheet so you need to remember it
How does the discriminant of a quadratic function affect its graph and roots?
- The discriminant tells us about the roots (or solutions) of the equation
- It also tells us about the graph of
- If Δ > 0 then and are two distinct values
- The equation has unequal real roots
- i.e. there are two distinct real solutions
- The graph of crosses the x-axis twice
- The equation has unequal real roots
- If then and are both zero
- The equation has equal real roots
- i.e. it has one repeated real solution
- The graph of touches the x-axis at exactly one point
- This means that the x-axis is a tangent to the graph
- The equation has equal real roots
- If then and are both undefined
- The roots of the equation are not real
- i.e. it has no real solutions
- The graph of never touches the x-axis
- This means that graph is wholly above (or below) the x-axis
- The roots of the equation are not real
How do I solve problems using the discriminant?
- Often at least one of the coefficients of a quadratic will be given as an unknown
- For example the letter may be used for the unknown constant
- You will be given a fact about the quadratic such as:
- The number of real solutions of the equation
- The number of roots (i.e. x-intercepts) of the graph
- To find the value or range of values of
- Find an expression for the discriminant
- Use
- Decide whether , or
- If the question says there are real roots but does not specify how many then use
- Solve the resulting equation or inequality for
- Find an expression for the discriminant
Exam Tip
- Questions won't always use the word discriminant
- It is important to recognise when its use is required
- Look for
- a number of roots or solutions being stated
- whether and/or how often the graph of a quadratic function intercepts the -axis
Worked example
A function is given by , where is a constant. The graph of intercepts the -axis at two different points.
a)
Show that .
The question says the graph 'intercepts the x-axis at two different points'
This means that the discriminant is greater than zero
Here , , and
Expand the brackets and collect terms
b)
Hence find the range of possible values of .
Solve the inequality, beginning by factorising
This tells us the graph of intercepts the horizontal axis at and
It can be helpful to sketch a graph here
will be true to the left of 0 and to the right of
Write these down as inequalities