Discriminants (Cambridge International IGCSE Further Maths)

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

$ax^{2}+bx+c=0$

• It is sometimes denoted by the Greek letter $Δ$ (capital delta)

Worked Example Find, in terms of the constant $k$, the discriminant of the quadratic equation $3x^{2}+2kx-k=kx^{2}-4kx+2$.

First write the quadratic equation in the form $ax^{2}+bx+c=0$.

$\begin{gathered} 3 x^{2} - k x^{2} + 2 k x + 4 k x - k - 2 & = 0 \\ ( 3 - k ) x^{2} + 6 k x - ( k + 2 ) & = 0 \end{gathered}$

It can be easier/clearer to pick out $a,b$ and $c$ first, before finding the discriminant.

$\begin{array}{rcl} a = 3 - k \\ b = 6 k \\ c = - ( k + 2 ) \end{array}$

The discriminant is $b^{2}-4ac$.

$\begin{gathered} \Delta = ( 6 k )^{2} - 4 \times ( 3 - k ) \times - ( k + 2 ) \\ \Delta = 36 k^{2} + 4 ( 6 + k - k^{2} ) \end{gathered}$

$\Delta=32k^{2}+4k+24$

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, $y = \mathrm{f} ( x )$ and $y = \mathrm{g} ( x )$ where

  • $\mathrm{f} ( x )$ is quadratic
  • $\mathrm{g} ( x )$ is linear

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

• STEP 1

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

• STEP 2

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

• STEP 3

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

    • 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 ($\mathrm{g} ( x )$) is a tangent to the curve ($\mathrm{f} ( x )$)
    • if $b^{2}-4ac<0$ (no real roots) the graphs do not intersect

Worked Example Show that the line with equation $y=3x-7$ is tangent to the curve with equation $y = ( x + 3 ) ( x - 2 )$.

STEP 1 - set the equations of the line and curve equal to each other.

$3 x - 7 = ( x + 3 ) ( x - 2 )$

STEP 2 - rearrange to quadratic form.

$\begin{gathered} 3 x - 7 & = x^{2} + x - 6 \\ x^{2} - 2 x + 1 & = 0 \end{gathered}$

STEP 3 - find the discriminant and interpret its value.

$\Delta = ( - 2 )^{2} - 4 ( 1 ) ( 1 ) = 0$

Final answer: Since the discriminant is zero, the line and the curve intersect at one point only. Therefore the line is a tangent to the curve.