Discriminants

Exam code: 4037

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

The discriminant is the part of the quadratic formula that is under the square root sign
It is $b^{2} - 4 a c$
- The quadratic formula, in full, is

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

where the quadratic equation is written in the form

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

Worked example

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

First write the quadratic equation in the form $a x^{2} + b x + c = 0$ .

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

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 $b^{2} - 4 a c$ .

$\begin{array}{rcl} ∆ & = & {(6 k)}^{2} - 4 \times (3 - k) \times - (k + 2) \\ ∆ & = & 36 k^{2} + 4 (6 + k - k^{2}) \end{array}$

$∆ = 32 k^{2} + 4 k + 24$

Applications of discriminant

If $b^{2} - 4 a c > 0$ the quadratic equation has two different real roots
- The graph of the quadratic will intersect the $x$ -axis twice (at the roots)
If $b^{2} - 4 a c = 0$ 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 $b^{2} - 4 a c < 0$ the quadratic equation has no (real) roots
- The graph of the quadratic will not intercept the $x$ -axis

For the graphs of two functions, $y = f (x)$ and $y = g (x)$ where
- $f (x)$ is quadratic
- $g (x)$ is linear

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

STEP 1
- Set $f (x) = g (x)$
STEP 2
- Rearrange into the form $h (x) = 0$ such that $h (x)$ is in the quadratic form $a x^{2} + b x + c = 0$
STEP 3
- Find the discriminant and thus determine the number of intersections between the graphs of $y = f (x)$ and $y = g (x)$
  - if $b^{2} - 4 a c > 0$ (two real roots) the graphs intersect twice
  - if $b^{2} - 4 a c = 0$ (equal roots) the graphs intersect once
  - this means the line ( $g (x)$ ) is a tangent to the curve ( $f (x)$ )
  - if $b^{2} - 4 a c < 0$ (no real roots) the graphs do not intersect

Worked example

Show that the line with equation $y = 3 x - 7$ 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.

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

STEP 3 - find the discriminant and interpret its value.

$∆ = {(- 2)}^{2} - 4 (1) (1) = 0$

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