GMAT PS : Quadratic Equation : Parabola cutting x-axis

Here is a question testing concepts of Quadratic equation and nature of roots of quadratic equation.

If y = x² + dx + 9 does not cut the x-axis, then which of the following could be a possible value of d?
I. 0
II. -3
III. 9

A. III only
B. II only
C. I and II only
D. II and III only
E. I and III only

Correct Answer : Choice C. Values that ‘d’ could take are 0 or -3

Explanation
The question states that the curve (parabola) does not cut the x-axis.

If y = x² + dx + 9 cuts the x-axis then, the points at which it cuts the x-axis will be the roots of the quadratic equation x² + dx + 9 = 0.

As any point on the x-axis will be a value on the number line, the roots will be real numbers.

However, if the curve does not cut the x-axis, then roots of the quadratic equation will be imaginary.

For a quadratic equation “ax² + bx + c = 0 to have imaginary roots, the discriminant b² – 4*a*c < 0 (the discriminant should be negative).

In this equation, d² – 36 < 0
Or d² < 36
i.e., -6 < d < 6.

Amongst the values given, d = 0 and d = -3 lie in this range.
Hence, choice C.