Determine m if the function f has two real roots. The roots are not equal. f = mx^2 + (m-1)*x - m + 2

The function f(x) has 2 distinct real roots if and only if
the discriminant delta is strictly positive.

delta = b^2 -
4ac

We'll identify a, b and
c.

a = m

b = m -
1

c = 2 - m

delta = (m-1)^2 -
4*m*(2 - m)

We'll impose the constraint: delta >
0

(m-1)^2 - 4*m*(2 - m) >
0

We'll expand the square and remove the
brackets:

m^2 - 2m + 1 - 8m + 4m^2 >
0

We'll combine like
terms:

5m^2 - 10m +
1>0

Now, we'll determine the roots of the expression
5m^2 - 10m + 1.

5m^2 - 10m + 1 =
0

We'll apply the quadratic
formula:

m1 =
[10+sqrt(100-20)]/10

m1 =
(10+4sqrt5)/10

m1 =
(5+2sqrt5)/5

m2 =
(5-2sqrt5)/5

The expression is positive
outside the roots, namely when m belongs to the
intervals:

(-infinite ,
(5-2sqrt5)/5) U ((5+2sqrt5)/5 , +infinite).