The discriminant

Notes

The quadratic formula tells us that the solutions to

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

are given by

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

But the $\sqrt{b^{2} - 4 a c}$ can cause problems. Sometimes $b^{2} - 4 a c < 0$ and so we can't find the square root (as a real number). Thus, we say that the quadratic has no real solutions.

The number

Δ = b^{2} - 4 a c

is called the discriminant and it tells us how many real roots a quadratic has.

⭐ Remember

$Δ = b^{2} - 4 a c$ is called the discriminant.

If $Δ > 0$ then the quadratic has two distinct, real roots.

If $Δ = 0$ then the quadratic has one real root.

If $Δ < 0$ then the quadratic has no real roots.

parabolas with two, one or no roots

Example

Find the number of real solutions to the equation
$x^{2} + 4 x = 21$
Find the number of real roots of the equation
$4 (15 x - x^{2}) = 225$
How many real values of $n$ satisfy
$- 5 - n - \frac{n^{2}}{2} = 0$
Given that the equation
$k x + k - 1 + x^{2} = 0$
has a repeated root, find the value of $k$ .

The discriminant ​

Notes ​

Example ​

Exercise ​

The discriminant

Notes

Example

Exercise