Question

If the quadratic equation $a{x}^2+bx+6=0$ does not have distinct real roots, then the least value of $2a+b$ is

• A. $2$
• B. $-3$
• C. $-6$
• D. $1$

Answer 1

The correct option is B $-3$

It is given that the expression $a{x}^2+bx+6=0$ doesn't have distinct real roots.

$\Rightarrow$ The graph of the function doesn't cross the x-axis.

$\Rightarrow$ The expression $f\left(x\right)=a{x}^2+bx+6$ is always greater than or equal to zero.

Hence, $f\left(x\right)\ge 0$.

$\Rightarrow$ $f\left(2\right)=a{\left(2\right)}^2+b\left(2\right)+6\ge 0$

$\Rightarrow$ $4a+2b+6\ge 0$

$\Rightarrow$ $2a+b+3\ge 0$

$\Rightarrow$ $2a+b\ge -3$