Question

If b>a ,then the equation (x-a)(x-b)-1=0 has it's roots in the interval ? please explain the method to find the intervals

Answer 1

One root will be interval `(-oo,a)` and other root will be in interval `b,oo)`

Explanation:

`(x-a)(x-b)-1=0` can be written as

`x^2-(a+b)x+ab-1=0`

then discriminant is `(a+b)^2-4(ab-1)=(a-b)^2+4>0`, it has two real roots

Further `f(a)=-1` and `f(b)=-1`, but `b>a` i.e. `a` and `b` are distinct as coefficient of `x^2` is positive (it is `1`), minima of `f(x)` is between `a` and `b`.

Hence one root will be interval `(-oo,a)` and other root will be in interval `(b,oo)`