The Discriminant of a Quadratic Equation Part 1 Recall that the quadratic equation ax2 + bx + c = 0 can be solved using the quadratic formula: For more information on the quadratic formula, see the following posts: Quadratic Formula – Part 1 Quadratic Formula – Part 2 Before we go on, you may also want to review the information on square roots that can be found here: Square Root Basics The discriminant of the quadratic equation ax2 + bx + c = 0 is the quantity Δ defined by Δ = b2 – 4ac In other words, the discriminant is simply the expression that appears under the square root in the quadratic formula. Although computing the discriminant of a quadratic equation does not give the roots (solutions) of the equation, it does give us a lot of information about the nature of the roots and the graph of the equation. For example, if the discriminant of the quadratic equation ax2 + bx + c = 0 is 0, then the quadratic formula simplifies to x = –b/2a, and we see that there is just one solution. If the coefficients a and b are integers, then the unique solution will be a rational number. Graphically, this means that the graph of the function y = ax2 + bx + c is a parabola that intersects the x-axis at one point. An example is given by the red parabola in the image above. Note: The discriminant does not tell us if the parabola opens upwards or downwards. However, this is easy to determine simply by looking at the value of a. If a > 0 (i.e. a is a positive number), then the parabola opens upwards. If a < 0 (i.e. a is a negative number), then the parabola opens downwards. So, the red parabola in the image above has an equation of the form y = ax2 + bx + c where a < 0 and Δ = 0 Example: Find the discriminant of x2 + 6x + 9 = 0. Then describe the nature of the roots of the equation, and describe the graph of the function y = x2 + 6x + 9. I will post a solution to this problem tomorrow, and then discuss other possibilities for the determinant. Feel free to post your own solutions in the comments. If you liked this article, please share it with your Facebook friends: Comments comments