Question

If $b>a$, then the equation $(x-a)(x-b)-1=0$ has

• A. both roots in $[a,b]$
• B. both roots in $(-\infty ,a)$
• C. both roots in $(b,\infty )$
• D. one root in $(-\infty ,a)$ and other in $(b,\infty )$

Answer 1

The correct option is D one root in $(-\infty ,a)$ and other in $(b,\infty )$

Let $f\left(x\right)=(x-a)(x-b)-1$
$f\left(a\right)=-1$ which is negative.
$f\left(b\right)=-1$ this is also negative.
Therefore, roots do not lie between $a$ and $b$.
Thus, one root in $(-\infty ,a)$ and other root lies in $(b,\infty )$.
Hence, option 'D' is correct.