Question

How do you find the discriminant and how many solutions does `h(x)=x^2-2x-35` have?

Answer 1

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 `h(x) = x^2-2x-35`

`Delta = (-2)^2 -4(1)(-35)`
`= 4+140 = 144`
`>0`
so there are two Real solutions