2.2.5 Discriminants

Discriminants

The discriminant of a quadratic is denoted by the Greek letter Δ (upper case delta)
For the quadratic function the discriminant is given by
- $Δ = b^{2} - 4 a c$
  - This is given in the formula booklet
The discriminant is the expression that is square rooted in the quadratic formula

If Δ > 0 then $\sqrt{b^{2} - 4 a c}$ and $- \sqrt{b^{2} - 4 a c}$ are two distinct values
- The equation $a x^{2} + b x + c = 0$ has two distinct real solutions
- The graph of $y = a x^{2} + b x + c$ has two distinct real roots
  - This means the graph crosses the x-axis twice
If Δ = 0 then $\sqrt{b^{2} - 4 a c}$ and $- \sqrt{b^{2} - 4 a c}$ are both zero
- The equation $a x^{2} + b x + c = 0$ has one repeated real solution
- The graph of $y = a x^{2} + b x + c$ 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 $\sqrt{b^{2} - 4 a c}$ and $- \sqrt{b^{2} - 4 a c}$ are both undefined
- The equation $a x^{2} + b x + c = 0$ has no real solutions
- The graph of $y = a x^{2} + b x + c$ 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

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 $Δ = b^{2} - 4 a c$
- 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

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 $x$ -axis
Be careful setting up inequalities that concern "two real roots" ( $∆ \geq 0$ ) as opposed to "two real distinct roots" ( $∆ > 0$ )

A function is given by $f (x) = 2 k x^{2} + k x - k + 2$ , where $k$ is a constant. The graph of $y = f (x)$ has two distinct real roots.

Show that

9 k^{2} - 16 k > 0

2-2-5-ib-aa-sl-discriminant-a-we-solution

Hence find the set of possible values of

k

2-2-5-ib-aa-sl-discriminant-b-we-solution