Question

If the roots of the equation, $x^2+6x+b=0$ are real and distinct and they differ by at most $8$, then $b$ lies in the interval

• A. $(7,25)$
• B. $(-7,9]$
• C. $[-7,9)$
• D. $[-7,16)$

Answer 1

The correct option is C $[-7,9)$

$x^2+6x+b=0$
Let the roots be $\alpha ,\beta$
$\therefore \alpha +\beta =-6$ and $\alpha \beta =b$

Given, $|\alpha -\beta |\le 8$
$\Rightarrow {|\alpha -\beta |}^2\le 8^2$
$\Rightarrow (\alpha +\beta {)}^2-4\alpha \beta \le 64$
$\Rightarrow 36-4b\le 64$
$\Rightarrow b\ge -7\cdots \left(1\right)$

Now, the roots are real and distinct.
$\Rightarrow D>0\Rightarrow 36-4b>0\Rightarrow b<9\cdots \left(2\right)$
Using $\left(1\right)$ and $\left(2\right)$,
$b\in [-7,9)$