Question

which statement must be true if a parabola represented by the equation $y = ax^2 + bx + c$ does not intersect the $x$-axis?

(1) $b^2 - 4ac = 0$

(2) $b^2 - 4ac < 0$

(3) $b^2 - 4ac > 0$, and $b^2 - 4ac$ is a perfect square.

(4) $b^2 - 4ac > 0$, and $b^2 - 4ac$ is not a perfect square.

Explanation:

Step1: Recall discriminant purpose

The discriminant $\Delta = b^2-4ac$ of $y=ax^2+bx+c$ tells the number of x-intercepts.

Step2: Match discriminant to no intercepts

If a parabola has no x-intercepts, it has no real roots, so $\Delta < 0$.

Step3: Evaluate each option

(1) $\Delta=0$ means one x-intercept; (3)(4) $\Delta>0$ means two x-intercepts; only (2) fits.

Answer:

(2) $b^2 - 4ac < 0$