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Solving Quadratic Equations:
     Solving by Graphing (page 5 of 6)

Sections: Solving by: factoring, taking roots, completing the square, using the formula, graphing

To be honest, solving "by graphing" is an achingly trendy but somewhat bogus topic. The basic idea behind solving by graphing is that, since the "solutions" to "ax² + bx + c = 0" are the x-intercepts of "y = ax² + bx + c", you can look at the x-intercepts of the graph to find the solutions to the equation. There are difficulties with "solving" this way, though.

When you graph a straight line like "y = 2x + 3", you can find the x-intercept (to a certain degree of accuracy) by drawing a really neat axis system, plotting a couple points, grabbing your ruler and drawing a nice straight line, and reading the (approximate) answer from the graph with a fair degree of confidence.

On the other hand, a quadratic graphs as a wiggly parabola. If you plot a few non-x-intercept points and then draw a curvy line through them, how do you know if you got the x-intercepts even close to being correct? You don't. The only way you can be sure of your x-intercepts is to set the quadratic equal to zero and solve. But the whole point of this topic is that they don't want you to do the (exact) algebraic solving; they want you to guess from the pretty pictures.

So "solving by graphing" tends to be neither "solving" nor "graphing". That is, you don't actually graph anything, and you don't actually do any of the "solving". Instead, you are told to punch some buttons on your graphing calculator and look at the pretty picture, and then you're told which other buttons to hit so the software can compute the intercepts. I think they're trying to teach the connection between x-intercepts and solutions, but the concept tends to get lost in all the button-pushing. Anyway...

To "solve" by graphing, either the book will give you a very neat, probably labelled, graph, and have you pick the points on the graph that represent solutions, or else you will be using your graphing calculator. Since different calculator models have different key-sequences, I cannot give instruction on the latter, so I will only give a couple examples of the former.

• Solve x² – 8x + 15 = 0 by using the following graph.

The point here is to look at the picture (hoping that the points really do cross at whole numbers, as it appears), and read the x-intercepts (and hence the solutions) from the picture.

The solution is x = 3, 5. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Since x² – 8x + 15 factors as (x – 3)(x – 5), we know that our answer is correct.

• Solve 0.3x² – 0.5x –  ⁵/₃ = 0 by using the following graph.

In this case, they labelled a bunch of points. Partly, this was to be helpful, because the x-intercepts are messy (so we could not have guessed them from the picture without the labels), but partly this was to confuse us, in the event that we had forgotten that only the x-intercepts, not vertices or other points, correspond to solutions.

The x-values of the two points where the line crosses the  x-axis are the solutions to the equation.

The solution is  x = ^–5/₃, ¹⁰/₃.

• Find the solutions to the following quadratic:

In this case, they haven't even given us the quadratic, so we can't even check our work algebraically.  (And, technically, they haven't even given us a quadratic to solve; they have only given us the picture of a parabola from which we are to approximate x-intercepts, which really is a different question.)

Read the "solutions" (the x-values of the points where the line crosses the x-axis) from the picture:

The solution is  x = –5.39, 2.76.

"Solving" quadratics by graphing is silly in "real life", and requires that the solutions be the simple factoring-type solutions (like "x = 3", rather than "x = –4 + sqrt(7)") that you could have found easily anyway. About the only advantage to covering this topic is if you learn the connection between solutions and x-intercepts. That is, if you learn and remember that the solutions to "polynomial equals zero" correspond to the x-intercepts of "y equals polynomial", then you can use your graphing calculator or other graphing software to help you solve general polynomials, if they happen not to factor (and many won't, in "real life"). But, in practice, given a quadratic to solve, you should not start by drawing a graph.

Which begs the question: When given a quadratic to solve, which of the methods should you use?

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