# 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

Nov 29, 2017

One root will be interval $\left(- \infty , a\right)$ and other root will be in interval b,oo)

#### Explanation:

$\left(x - a\right) \left(x - b\right) - 1 = 0$ can be written as

${x}^{2} - \left(a + b\right) x + a b - 1 = 0$

then discriminant is ${\left(a + b\right)}^{2} - 4 \left(a b - 1\right) = {\left(a - b\right)}^{2} + 4 > 0$, it has two real roots

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

Hence one root will be interval $\left(- \infty , a\right)$ and other root will be in interval $\left(b , \infty\right)$