Discriminants
What is the discriminant of a quadratic function?
- The discriminant of a quadratic is denoted by the Greek letter Δ (upper case delta)
- For the quadratic function the discriminant is given by
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- This is given in the formula booklet
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- The discriminant is the expression that is square rooted in the quadratic formula
How does the discriminant of a quadratic function affect its graph and roots?
- If Δ > 0 then and are two distinct values
- The equation has two distinct real solutions
- The graph of has two distinct real roots
- This means the graph crosses the x-axis twice
- If Δ = 0 then and are both zero
- The equation has one repeated real solution
- The graph of has one repeated real root
- This means the graph touches the x-axis at exactly one point
- This means that the x-axis is a tangent to the graph
- If Δ < 0 then and are both undefined
- The equation has no real solutions
- The graph of has no real roots
- This means the graph never touches the x-axis
- This means that graph is wholly above (or below) the x-axis
Forming equations and inequalities using the discriminant
- Often at least one of the coefficients of a quadratic is unknown
- Questions usually use the letter k for the unknown constant
- You will be given a fact about the quadratic such as:
- The number of solutions of the equation
- The number of roots of the graph
- To find the value or range of values of k
- Find an expression for the discriminant
- Use
- Decide whether Δ > 0, Δ = 0 or Δ < 0
- If the question says there are real roots but does not specify how many then use Δ ≥ 0
- Solve the resulting equation or inequality
- Find an expression for the discriminant
Exam Tip
- Questions will rarely use the word discriminant so 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
- Look for
- Be careful setting up inequalities that concern "two real roots" () as opposed to "two real distinct roots" ()
Worked Example
A function is given by , where is a constant. The graph of has two distinct real roots.
a)
Show that .
b)
Hence find the set of possible values of .