Question

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

Answer 1

The discriminant of `x^2-2=0` is 8,
which means there are 2 Real solutions to this equation.

Explanation:

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

`Delta {(< 0, rarr "there are no Real solutions"), (= 0, rarr "there is exactly 1 Real solution"), (> 0, rarr "there are 2 Real solutions") :}`

Converting the given equation `x^2 -2 = 0`
into standard form
`color(white)("XXXX")` `1x^2 +0x -2 = 0`
gives us
`color(white)("XXXX")` `a=1` `color(white)("XXXX")` `b=0` `color(white)("XXXX")` `c=-2`

So the discriminant is
`color(white)("XXXX")` `Delta = 0^2 - 4(1)(-2) = +8`

which implies that there are 2 Real solutions for `x`