Question

What is the discriminant of `x^2+2x+8=0` and what does that mean?

Answer 1

The discriminant of `x^2+2x+8 = 0` is `(-28)` which means that this equation has no Real solutions.

Explanation:

For a quadratic equation in the form
`color(white)("XXXX")` `ax^2+bx+c=0`
the discriminant is
`color(white)("XXXX")` `Delta = b^2-4ac`

The discriminant is the portion of the quadratic formula for solving a quadratic equation:
`color(white)("XXXX")` `x = (-b+-sqrt(b^2-4ac))/(2a)`

Seen in this context, it should be clear why:
`color(white)("XXXX")` #Delta { ( > 0, rarr, 2 " Real solutions"), (=0,rarr, 1 " Real solution"), (< 0, rarr, " no Real solutions"):}#

For the given quadratic
`color(white)("XXXX")` `x^2+2x+8 = 0`
the discriminant is
`color(white)("XXXX")` `Delta = 2^2 - 4(1)(8) = -28`
which tells us that this equation has no Real solutions.