In which cases do quadratic functions have no real roots?

Quadratic functions can be expressed in the standard form ( ax^2 + bx + c = 0 ). The roots of this equation can be determined using the discriminant ( D ), which is calculated as ( D = b^2 - 4ac ).

When the discriminant is negative, it indicates that there are no real solutions to the quadratic equation. This is because a negative discriminant implies that the expression under the square root in the quadratic formula ( x = \frac{-b \pm \sqrt{D}}{2a} ) results in taking the square root of a negative number, which does not yield any real values. Instead, the roots will be complex or imaginary numbers.

Therefore, a quadratic function has no real roots specifically when the discriminant is negative.

In contrast, a discriminant of zero indicates that there is exactly one real root (the vertex of the parabola touches the x-axis), and a positive discriminant signifies two distinct real roots (the parabola intersects the x-axis at two points). The case where all the coefficients are zero would not typically yield a proper quadratic function, leading to an entirely different consideration of solutions.