Let us consider the following quadratic function: f(x) = -x2 + 15x + 20

The solutions of the quadratic equation -x2 + 15x + 20 = 0 correspond to the zeros or the roots of the function f(x) = -x2 + 15x + 20.
These are the points where the graph of f(x) cuts the x-axis. The graph cuts the Y-axis at 20.

### The Discriminant and Roots of the Quadratic Equation -x2 + 15x + 20 = 0

The standard form of a quadratic equation is ax2 + bx + c = 0, where "a" does not equal 0. Note that if a = 0, the x2 term would disappear and the equation would be linear.
Looking at the given quadratic function  a = -1, b = 15, c = 20.
The discriminant D = b2 - 4ac = 152 - 4 * (-1) * (20) = 305.0
The roots of the equation are  (-b - √D)/2a and (-b + √D)/2a
= (-(15) - √305.0)/(2(-1)) and (-(15) + √305.0)/(2(-1))
16.232 and -1.232

The discriminant is positive. Hence, the roots are real and unequal. The quadratic curve cuts the X axis at two distinct points.
The value of 'x' in the intervals (x < 16.232) and (x > -1.232) satisfy the inequality -x2 + 15x + 20 < 0

Graph of y = f(x) = -x2 + 15x + 20 ### Geometric and Graphical interpretation: Curve Sketching

The function f(x) = -x2 + 15x + 20 is the quadratic function. The graph of any quadratic function has the same general shape. This shape is called a parabola. The location and size of the parabola, and how it opens, depend on the values of coefficients in the function. This parabola has a maxima point and opens downwards. The graph of the parabola is symmetric with respect to the vertical line passing through the vertex.

The x-coordinate of the vertex will be located at x = (-b/2a) = (-(15))/(2*-1) = 7.5, and the y-coordinate of the vertex is 76.25 which we obtain by substituting the value x = 7.5 in -x2 + 15x + 20. This is the maxima value attained by the quadratic function f(x). The derivative of the function is 0 at this point. This point is a turning point or a stationary point.
The solutions of the quadratic equation -x2 + 15x + 20 = 0 correspond to the zeros or the roots of the function f(x) = -x2 + 15x + 20.  As shown in the figure, when the coefficients a, b, and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x-coordinates of the points where the graph touches the x-axis. The discriminant is positive. Hence, the roots are real and unequal. The quadratic curve cuts the X axis at two distinct points.
The value of 'x' in the intervals (x < 16.232) and (x > -1.232) satisfy the inequality -x2 + 15x + 20 < 0

The vertex is the lowest point on the parabola if the parabola opens upward (coefficient a > 0) and is the highest point on the parabola if the parabola opens downward (coefficient a < 0)

Check the plot of another quadratic curve here.