Question

What is the discriminant of `x^2-4=0` and what does that mean?

Answer 1

The discriminant is 8. It tells you that there are two separate real roots to the equation.

Explanation:

If you have a quadratic equation of the form

`ax^2+bx+c=0`

The solution is

`x = (-b±sqrt(b^2-4ac))/(2a)`

The discriminant `Δ` is `b^2 -4ac`.

The discriminant "discriminates" the nature of the roots.

There are three possibilities.

• If `Δ > 0`, there are two separate real roots.
• If `Δ = 0`, there are two identical real roots.
• If `Δ <0`, there are no real roots, but there are two complex roots.

Your equation is

`x^2 - 2 = 0`

`Δ = b^2 – 4ac = (0)^2 -4×1×(-2) = 0 +8 = 8`

This tells you that there are two separate real roots.

We can see this if we solve the equation.

`x^2 -2 = 0`

`x = (-b±sqrt(b^2-4ac))/(2a) = (-0±sqrt((0)^2 -4×1×(-2)))/(2×1) = ±sqrt(0+8)/2 = ±sqrt8/2 = ±(2sqrt2)/2 = ±sqrt2`#

`x = sqrt2` and `x = -sqrt2`

There are two separate real roots to the equation.