The equation x^2 + kx + 9 = 0, where k is a constant, has no real roots. Which of the following gives the complete set of possible values of k?

A Level Maths 9709: Discriminant Condition for No Real Roots

Compare $x^2 + kx + 9$ with the general form $ax^2 + bx + c$ :

$a = 1, \qquad b = k, \qquad c = 9$

A quadratic $ax^2+bx+c=0$ has no real roots exactly when its discriminant is negative:

$b^2 - 4ac < 0$

$k^2 - 4(1)(9) < 0$

$k^2 - 36 < 0$

$k^2 < 36$

Taking square roots and remembering both signs:

$-6 < k < 6$

B ( $k<-6$ or $k>6$ ): this is where $k^2-36>0$ , the condition for two distinct real roots, not none. The inequality has been reversed.
C ( $-3<k<3$ ): comes from dropping the factor $4$ in $4ac$ , mistakenly solving $k^2<9$ (using $ac=9$ directly) instead of the correct $k^2<4(9)=36$ .
D ( $k<6$ ): keeps only the upper bound from $k^2<36$ and forgets that $k$ can also be any value greater than $-6$ , i.e. the lower bound $k>-6$ is missing.