Question

If the roots of the quadratic equation $x^2+6x+b=0$ are real and distinct and they differ by at most $4,$ then the range of $b$ is

• A. $[-3,5]$
• B. $[5,9)$
• C. $[6,10]$
• D. $[5,\infty ]$

Answer 1

The correct option is B $[5,9)$

Let $\alpha$ and $\beta$ be the roots of quadratic equation $x^2+6x+b=0.$ Then
$0<|\alpha -\beta |\le 4\Rightarrow 0<|\alpha -\beta {|}^2\le 16\Rightarrow 0<(\alpha +\beta {)}^2-4\alpha \beta \le 16\Rightarrow 0<36-4b\le 16\Rightarrow 9>b\ge 5\dots (i)$

Also roots are real and distinct.
$\Rightarrow D>0\Rightarrow 36-4b>0\Rightarrow 9>b\dots \left(ii\right)$

From $\left(i\right)$ and $\left(ii\right),$ we get
$b\in [5,9)$