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

1 Answer
Nov 29, 2017

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)#