Discriminants (DP IB Analysis & Approaches (AA): HL): Revision Note

Discriminants

What is the discriminant of a quadratic function?

• The discriminant of a quadratic is denoted by the Greek letter $\mathrm{\Delta }$ (upper case delta)

• For the quadratic function $a{x}^2+bx+c$ the discriminant is given by

  • $\mathrm{\Delta }=b^2-4ac$ 

    • This is given in the formula booklet

• 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 $\mathrm{\Delta }>0$ then $\sqrt{b^2-4ac}$ and $-\sqrt{b^2-4ac}$ are two distinct values

  • The equation $a{x}^2+bx+c=0$ has two distinct real solutions

  • The graph of $y=a{x}^2+bx+c$ has two distinct real roots

    • This means the graph crosses the $x$-axis twice

• If $\mathrm{\Delta }=0$ then $\sqrt{b^2-4ac}$ and $-\sqrt{b^2-4ac}$ are both zero

  • The equation $a{x}^2+bx+c=0$ has one repeated real solution

  • The graph of $y=a{x}^2+bx+c$ 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 $\mathrm{\Delta }<0$ then $\sqrt{b^2-4ac}$ and $-\sqrt{b^2-4ac}$ are both undefined

  • The equation $a{x}^2+bx+c=0$ has no real solutions

  • The graph of $y=a{x}^2+bx+c$ has no real roots

    • This means the graph never touches the x-axis

    • This means that graph is wholly above (or wholly below) the $x$-axis

Forming equations and inequalities using the discriminant

• A common question is to find an unknown coefficient in a quadratic

  • 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 using $\mathrm{\Delta }=b^2-4ac$

  • Decide whether $\mathrm{\Delta }>0$, $\mathrm{\Delta }=0$ or $\mathrm{\Delta }<0$

    • If the question says there are real roots but does not specify how many then use $\mathrm{\Delta }\ge 0$

  • Solve the resulting equation or inequality to find the value(s) of $k$