These three possibilities for any real number -- positive, negative, or 0 -- is called the Law of Trichotomy.
The quadratic will be positive -- above the x-axis -- for values of x to the left and right of the roots.
If you have a programmable or graphing calculator, it will most likely have a built-in program to find the roots of polynomials.
Only one of the roots is real, all the other six roots contain the symbol i, and are thus complex roots (more about those later on).
When we are asked to solve a quadratic equation, we are really being asked to find the roots.
If the discriminant of a quadratic function is less than zero, that function has no real roots, and the parabola it represents does not intersect the x-axis.


If the discriminant of a quadratic function is equal to zero, that function has exactly one real root and crosses the x-axis at a single point. If the discriminant of a quadratic function is greater than zero, that function has two real roots (x-intercepts). We have already seen that completing the square is a useful method to solve quadratic equations. Since the quadratic formula requires taking the square root of the discriminant, a negative discriminant creates a problem because the square root of a negative number is not defined over the real line. Thus, a parabola has exactly one real root when the vertex of the parabola lies right on the x-axis.
This method can be used to derive the quadratic formula, which is used to solve quadratic equations.


The discriminant is important because it tells you how many roots a quadratic function has. In fact, as you will see shortly, , a polynomial of degree 4, has indeed only the two real roots -1 and 2.



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