if a `lt c lt b, ` then check the nature of roots of the equation
`(a -b)^(2) x^(2) + 2(a+ b - 2c)x + 1 = 0`

Answer

Roots are non-real .

Explanation

The discriminant of the given equation is
`D = 4 ( a + b - 2c)^(2) - 4(a - b)^(2)`
`= 4 (a + b - 2c - a b) ( a + b - 2c + a - b) `
` = 16 ( a - c) (b - c) lt 0 [because a lt c lt b]`
Hence, the roots of the given equation are complex.