The Quadratic Formula: The Discriminant and Graphs

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Here's the graph of the associated function, y = x² + 2x − 1:

The x-intercepts (that is, the solutions from above) are marked in red. These are the spots where the associated function, y, was equal to zero.

Note that the x-intercepts of the associated function match with the solutions to the original equation. And recall that the value of the discriminant (the part inside the square root in the Quadratic Formula) was positive.

This relationship between the value inside the square root (the discriminant), the type of solutions (two different solutions, one repeated solution, or no rea solutions), and the number of x-intercepts (on the corresponding graph) of the quadratic is summarized in the following table:

What is the point of the above table? The point is that there is a clear and consistent connection between solutions of quadratic equations (where you've got "(quadratic) = 0") and the graphs of the associated functions (which will be "y = (quadratic)"); namely, that the real solutions of the equation will be the x-intercepts of the graph.

So if you're wanting to check your solutions to a given quadratic equation, plug the associated quadratic function into your graphing calculator, and check the graph. Does the graph cross the x-axis at your solution points? Then you probably have the right answer. Did you get a repeated solution (that is, a soluton with zero inside the square root), but the graph doesn't touch the axis at all? Then your solution is probably wrong. And so forth. Use the graphs to check the solutions.

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Probably the most important thing to remember when using the Quadratic Formula (other than the Formula itself, which you should memorize) is that you must do each step clearly and completely, so you don't lose your denominators or plus-minuses or square roots. Don't skip stuff, and you should do fine.

I've been grading homework and tests for too many years to be kidding about this. Really, truly; you want to do your work neatly and completely every single time!