First slide

Theory of expressions

Question

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

Moderate

A

both roots in $[a, b]$
B

both roots in $(- \infty, a)$
C

roots in $(- \infty, a)$ and other in $(b, \infty)$
D

both roots in $(b, \infty)$

Solution

We have
$\begin{array}{l} x - a) (x - b) - 1 = 0 \\ x^{2} - x (a + b) + a b - 1 = 0 \end{array}$
bet the roots of this equation. Then,
$α + β = a + b$ and $α β = a b - 1$
$\Rightarrow$ If one root is less than a, then the other root is greatest than b.
$\Rightarrow$ One root in $(- \infty, a)$ and the other is in $(b, \infty)$
$A L I T E R$ Clearly, a and b are the roots of the equation $(x - a) (x - b) = 0$
Therefor 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 $(- \infty, a)$ and other in $(b, \infty)$ .

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