91Ó°ÊÓ

The discriminant is a special expression used with quadratic equations to reveal important information about their roots. For a general quadratic equation, written as \( ax^2 + bx + c = 0 \), the discriminant is given by the formula \( \Delta = b^2 - 4ac \). This value is crucial because it tells us whether the equation has real roots:

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root, sometimes called a repeated or double root.
• If \( \Delta < 0 \), there are no real roots, meaning the roots are complex or imaginary.

In the context of the function \( f(x) = A x^2 + B x + C \), with \( A > 0 \), if \( B^2 - 4AC \leq 0 \), the equation has at most one real root. This results in the quadratic function not intersecting or merely touching the x-axis. It supports the condition that the graph of the function remains above or on the x-axis, fulfilling the non-negative function requirement.

A non-negative function is one that does not take on any negative values for its entire domain. For the quadratic function \( f(x) = A x^2 + B x + C \), where \( A > 0 \), proving that it is non-negative for all \( x \) involves checking its relationship with the x-axis. The function is non-negative if its graph never dips below the x-axis.

For a parabola to be non-negative, it implies:

• The vertex of the parabola lies at or above the x-axis.
• The parabola opens upwards, which is already given because \( A > 0 \).
• The discriminant \( B^2 - 4AC \) must be less than or equal to zero, indicating at most one contact with the x-axis, either as a tangent or not at all.

By ensuring these conditions, the function \( f(x) \geq 0 \) holds true for all \( x \), meaning it is always non-negative.

The roots of a quadratic equation represent the values of \( x \) where the function equals zero, i.e., where \( ax^2 + bx + c = 0 \). For our specific quadratic function \( f(x) = A x^2 + B x + C \) with \( A > 0 \), determining the roots gives insight into where the function might cross the x-axis.

The roots are determined by the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The solutions derived from this formula depend heavily on the discriminant \( b^2 - 4ac \):

• If the discriminant is positive, there are two distinct roots, and the graph of the quadratic crosses the x-axis at two points.
• If the discriminant is zero, there's exactly one root, where the graph touches the x-axis at the vertex.
• If the discriminant is negative, there are no real roots, and the graph does not touch or cross the x-axis at all.

When \( B^2 - 4AC \leq 0 \) for our quadratic function, it confirms that the graph does not dip below the x-axis, satisfying the condition where the function remains non-negative.