Problem

An upward-opening real-coefficient parabola stays strictly above the \(x\text{-}\text{axis}\). Determine its number of \(x\text{-}\text{intercepts}\), infer the \(\text{discriminant}\)'s sign, and classify its \(\text{two}~\text{roots}\) over the complex numbers.

Prompt graph of an upward-opening parabola whose vertex and every point lie strictly above the x-axis. The curve never touches or crosses the x-axis; the intercept count and root classification are withheld.

Big Picture

What this problem is really about

The graph reveals real-root information through its intersections with the x-axis. We’ll inspect whether the parabola ever reaches zero, translate that intercept count into the discriminant’s sign, distinguish no real roots from no roots at all, and use real coefficients to infer a conjugate pair.

Answer choices

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It has 1 x-intercept and one repeated real root.

It has 0 x-intercepts and two nonreal complex conjugate roots.

It has 2 x-intercepts and two distinct real roots.

It has 0 x-intercepts and two nonreal roots that are not conjugates.

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An upward-opening real-coefficient parabola stays strictly above the \(x\text{-}\text{axis}\). Determine its number of \(x\text{-}\text{intercepts}\), infer the \(\text{discriminant}\)'s sign, and classify its \(\text{two}~\text{roots}\) over the complex numbers.

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