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 values of $b$ is:

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

Answer 1

Answer: (B)

$[5,9)$

Given quadratic equation is
$x^2+6x+b=0$
Since, roots are real and distinct i.e. $D>0$
$\Rightarrow 36-4b>0$
$\Rightarrow b<9$
Also, ${(\alpha -\beta )}^2={(\alpha +\beta )}^2-4\alpha \beta$
${(\alpha -\beta )}^2=36-4b$
Given, $\alpha -\beta \le 4$
$\Rightarrow {(\alpha -\beta )}^2\le 16$
$36-4b\le 16$
$\Rightarrow b\ge 5$
Hence, range of $b$ is $[5,9)$.