Question

How do you find the discriminant of `3x^2+2x-1=0` and use it to determine if the equation has one, two real or two imaginary roots?

Answer 1

There are 2 real roots
(see below)

Explanation:

For a quadratic with the general form:
`color(white)("XXX")color(red)ax^2+color(blue)bx+color(green)c=0`
the discriminant is:
`color(white)("XXX")color(orange)(Delta)=color(blue)b^2-4color(red)acolor(green)c`
with
`color(white)("XXX")color(orange)(Delta){(color(orange)( < 0) rarr 2" imaginary roots"),(color(orange)(= 0) rarr 1 " real root"),(color(orange)(> 0) rarr 2 " real roots"):}`

Given
`color(white)("XXX")color(red)3x^2+color(blue)(2)x+color(green)((-1))=0`
the discriminant is
`color(white)("XXX")color(orange)(Delta)=color(blue)2^2-4 * color(red)3 * color(green)((-1))=color(orange)(20)`
`rArr` there are 2 real roots.