If b>a, the the equation  (x−a)(x−b)−1=0 has

We havex−a)(x−b)−1=0x2−x(a+b)+ab−1=0 bet the roots of this equation. Then, α+β=a+b and αβ=ab−1⇒ If one root is less than a, then the other root is greatest than b.⇒One root in (−∞,a) and the other is in (b,∞)ALITER Clearly, a and b are the roots of the equation (x−a)(x−b)=0Therefor the curve y=(x−a)(x−b) opens upward and cuts x-axis at (a,0) and (b,0)The curve y=(x−a)(x−b)−1 is obtained by translating y=(x-a) (x-b) through one unit in vertically downward direction. So, it will cross x-axis at two points one lying on the left of (a, 0) and other one the right of (b, 0). Hence, one of the roots lies in (−∞,a) and other in (b,∞).