Question

How do you find the discriminant and how many and what type of solutions does `6x^2=13x` have?

Answer 1

Ripan's solutions are correct but don't answer the more general question.

Solutions to a quadratic of the form `ax^2+bx+c=0` are given by
the quadratic formula `(-b+-sqrt(b^2-4ac))/(2a)`

The sub-expression within the square root determines the number (and type) of solutions; this sub-expression is called the "discriminant" and is typically expressed as:
`Delta = b^2-4ac`
with the conditions
`Delta { (< 0 " there are no Real solutions"),(=0" there is 1 Real solution"),(>0" there are 2 Real solutions"):}`

Given `6x^2=13x`
we can re-arrange this into the general form
`6x^2-13x+0 = 0`

and
`Delta = (13)^2 -4(6)(0) = 169>0`
so `6x^2= 13x` has 2 Real solutions