Question

How to use the discriminant to find out what type of solutions the equation has for `3x^2 - x + 2 = 0`?

Answer 1

Zero roots

Explanation:

Quadratic formula is `x=(-b+-sqrt(b^2-4ac))/(2a)`
or
`x=-b/(2a)+-(sqrt(b^2-4ac))/(2a)`

We can see that the only part that matters is `+-(sqrt(b^2-4ac))/(2a)`
as if this is zero then it says that only the vertex `-b/(2a)` lies on the x-axis

We also know that `sqrt(-1)` is undefined as it doesn't exist so when `b^2-4ac=-ve` then the function is undefined at that point showing no roots

Whilst if `+-(sqrt(b^2-4ac))/(2a)` does exist then we know it is being plussed and minused from the vertex showing their are two roots

Summary:
`b^2-4ac=-ve` then no real roots
`b^2-4ac=0` one real root
`b^2-4ac=+ve` two real roots

So
`(-1)^2-4*3*2=1-24=-23` so it has zero roots