Understanding Quadratic Polynomials and Their Graphs

A quadratic polynomial is an expression of the form \(ax^2 + bx + c\), where \(a, b, c\) are constants and \(a \neq 0\). The graph of a quadratic polynomial is a parabola.

The roots of the quadratic polynomial are the values of \(x\) for which \(ax^2 + bx + c = 0\). Graphically, the real roots are the x-coordinates of the points where the parabola intersects the x-axis.

Graph Entirely Above the X-axis

If the graph of a quadratic polynomial lies entirely above the x-axis, it means that the parabola never touches or crosses the x-axis. This implies that there are no real values of \(x\) for which \(ax^2 + bx + c = 0\).

For the parabola to lie entirely above the x-axis, two conditions must be met:

1. The parabola must open upwards. This happens when the coefficient of \(x^2\), \(a\), is positive (\(a > 0\)). If \(a < 0\), the parabola opens downwards and would eventually go below the x-axis.
2. The parabola must not intersect the x-axis. This means the quadratic equation \(ax^2 + bx + c = 0\) has no real roots.

Nature of Roots and the Discriminant

The nature of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is determined by the discriminant, denoted by \(\Delta\). The discriminant is calculated as \(\Delta = b^2 - 4ac\).

There are three possibilities for the nature of the roots based on the value of the discriminant:

• If \(\Delta > 0\), there are two distinct real roots. The graph intersects the x-axis at two different points.
• If \(\Delta = 0\), there is exactly one real root (or two equal real roots). The graph touches the x-axis at exactly one point (the vertex lies on the x-axis).
• If \(\Delta < 0\), there are two complex (non-real) roots, which are conjugates of each other. The graph does not intersect the x-axis.

Connecting Graph Position to Root Nature

The question states that the graph of the quadratic polynomial lies entirely above the x-axis. As discussed, this means the parabola does not intersect the x-axis at any point. According to the relationship between the discriminant and the graph, this situation occurs precisely when the discriminant is negative (\(\Delta < 0\)).

When the discriminant \(\Delta < 0\), the quadratic equation \(ax^2 + bx + c = 0\) has no real roots. In this case, the roots are complex numbers.

Therefore, if the graph of a quadratic polynomial lies entirely above the x-axis, both of its roots must be complex.

Let's check the options based on this understanding:

• Option 1: Both the roots are real. This corresponds to \(\Delta \ge 0\), meaning the graph intersects or touches the x-axis. This is incorrect.
• Option 2: One root is real and the other is complex. This is not possible for a polynomial with real coefficients. Complex roots always appear in conjugate pairs. This is incorrect.
• Option 3: Both the roots are complex. This corresponds to \(\Delta < 0\), meaning the graph does not intersect the x-axis. This is correct.
• Option 4: Cannot say. Based on the position of the graph relative to the x-axis, we can definitively determine the nature of the roots. This is incorrect.

Thus, if the graph of a quadratic polynomial lies entirely above the x-axis, both the roots are complex.

Revision Table: Quadratic Polynomial Graphs and Roots

Additional Information: Parabola Opening Direction

For the graph of \(ax^2 + bx + c\) to lie entirely above the x-axis, it must open upwards (\(a > 0\)). If \(a < 0\), the parabola opens downwards. A downward-opening parabola with its vertex above the x-axis would still intersect the x-axis at two points as \(x \to \pm \infty\), the function value \(ax^2 + bx + c \to -\infty\).

So, the condition for the graph to be entirely above the x-axis requires both \(a > 0\) and \(\Delta < 0\). The condition \(\Delta < 0\) directly implies that the roots are complex, regardless of the sign of \(a\) (though the graph's position relative to the x-axis depends on \(a\)).

If the question were "If the graph lies entirely below the x-axis", then \(a\) would have to be negative (\(a < 0\)) and \(\Delta\) would still need to be negative (\(\Delta < 0\)). Again, the roots would be complex.