When we have a quadratic equation such as ax2+bx+c = 0, we can find out the value of x by plugging the coefficients into a formula to solve the equation. This formula is called the quadratic formula:
The quadratic formula is composed of a solution (x) and coefficients.
The coefficients are the values of a, b, and c. Below is a variation of the quadratic formula.
If you take a good look at both formulas, you will notice that the quadratic formula and its variation both contain the square root of b2-4ac. This is called the discriminant. The discriminant has the interesting property of being an indicator for how many real roots are contained in the solution.
Arithmetic precision and weather or not rescaling has occured affects the computational error of the quadratic formula. The web-based activity below was designed to aid in the exploration of finding the roots of a quadratic equation. The bottom window computes the roots using 25 digit precision and displays them rounded to the current selected precision. Compare the results of this bottom window with the results calculated in the current precision using both variations of the quadratic formula found on the right.
What do you notice about the results of each of the quadratic formulas using a low precision?
References: (Heath 17-18, Scientific Computing: An Introductory Survey)
Applet Source QuadForm.java,
Written by Nicholas Exner.
- How many solutions do we have when b2-4ac > 0?
- How many solutions do we have when b2-4ac = 0?
- How many solutions do we have when b2-4ac < 0?
- What conclusions can we draw when summarizing the above findings?