Discriminants (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Discriminants

What is a discriminant?

• The discriminant is the part of the quadratic formula that is under the square root sign 

• It is $b^2-4ac$

  • The quadratic formula, in full, is

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

where the quadratic equation is written in the form

$a{x}^2+bx+c=0$

• It is sometimes denoted by the Greek letter $\mathrm{\Delta }$ (capital delta)

Applications of discriminant

How does the value of the discriminant affect roots?

• There are three options for the outcome of the discriminant:

  • If $b^2-4ac>0$ the square root part of the quadratic formula can be calculated leading to two solutions (values of $x$)

    • i.e. two different real roots

  • If  $b^2-4ac=0$ the square root part of the quadratic formula will be zero leading to one solution

    • i.e. one repeated root or two equal roots

  • If  $b^2-4ac<0$ the square root part of the quadratic formula cannot be calculated leading to no solutions

    • i.e. no (real) roots

How do I sketch quadratic graphs using the discriminant?

• If $b^2-4ac>0$ the quadratic equation has two different real roots

  • The graph of the quadratic will intersect the $x$-axis twice (at the roots)

• If  $b^2-4ac=0$ the quadratic equation has two equal roots (one root)

  • The graph of the quadratic will intersect (touch) the $x$-axis once (at the root)

• If  $b^2-4ac<0$ the quadratic equation has no (real) roots

  • The graph of the quadratic will not intercept the $x$-axis

How do I use the discriminant to find the number of intersections between a line and a curve?

• For the graphs of two functions, $y=\mathrm{f}\left(x\right)$ and $y=\mathrm{g}\left(x\right)$ where

  • $\mathrm{f}\left(x\right)$ is quadratic

  • $\mathrm{g}\left(x\right)$ is linear

the number of intersections between the graphs can be found using the discriminant.

• STEP 1

  • Set $\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)$

• STEP 2

  • Rearrange into the form $\mathrm{h}\left(x\right)=0$ such that $\mathrm{h}\left(x\right)$ is in the quadratic form $a{x}^2+bx+c=0$

• STEP 3

  • Find the discriminant and thus determine the number of intersections between the graphs of $y=\mathrm{f}\left(x\right)$and $y=\mathrm{g}\left(x\right)$

    • if $b^2-4ac>0$ (two real roots) the graphs intersect twice

    • if $b^2-4ac=0$ (equal roots) the graphs intersect once

    • this means the line ($\mathrm{g}\left(x\right)$) is a tangent to the curve ($\mathrm{f}\left(x\right)$)

    • if $b^2-4ac<0$ (no real roots) the graphs do not intersect