Question

The entire graphs of the equation $y=x^2+kx-x+9$ is strictly above the x-axis if and only if

• A. k<7
• B. -5<k<7
• C. k>-5
• D. -7<k<5

Answer 1

The correct option is B

-5<k<7

$y=x2+(k-1)x+9={(x+\frac{k+1}{2})}^2+9-{\left(\frac{k-1}{2}\right)}^2$
For entire graph to be above x-axis, we should have
$9-{\left(\frac{x-1}{2}\right)}^2$ > 0
$\Rightarrow k^2-2k-35<0\Rightarrow (k-7)(k+5)$ < 0
$\Rightarrow$
i.e., -5<k<7