SOLUTION: Without solving the equation how can i determine the solutions to the 7-6x-x^2=0 using only the graph.Please explain how you obtain your answer by looking at the graph.Does this fu

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Question 229600: Without solving the equation how can i determine the solutions to the 7-6x-x^2=0 using only the graph.Please explain how you obtain your answer by looking at the graph.Does this function have a maximum or a minimumThank you so much in advance!
Found 2 solutions by Earlsdon, solver91311:
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Determine the solutions of7-6x-x%5E2+=+0 by graphing.
First, the solutions to this quadratic equation are found where the curve (a parabola) intersects the x-axis.
Second, if the coefficient of the x%5E2term is positive, the parabola opens upwards and the function has a minimum. If the coefficient is negative, as it is in this case, then the parabola opens downwards and the function has a maximum.
Let's look at the graph of this function:
graph%28400%2C400%2C-10%2C5%2C-5%2C20%2C7-6x-x%5E2%29
Notice that the parabola intersects the x-axis in two places so there are two solutions (also called roots). The maximum occurs at (-3, 16)
highlight%28x+=+-7%29 and highlight%28x+=+1%29

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


When you are finding the solutions of , what you are really doing is saying, "What value(s) of will make when . Technically, you didn't pose your question correctly. What you should have said was:

"...determine the solutions to the equation 7-6x-x^2=0 using only the graph of the function ."

The solutions to your equation from a graphical standpoint are those points on the graph of the form . That's because in order for to be a solution to , must equal zero. So where on the coordinate plane do you find points where the -coordinate is equal to zero? Answer: The -axis. In other words, the solutions to are the -coordinates of the points where the graph of crosses the -axis.

Practically speaking, this is a rather poor way to solve a quadratic equation. In the first place, you quite often have to solve in order to graph . Secondly, even if you use a computer program or graphing calculator to create your graph, you are only looking at an approximation of the answer that is as accurate and precise as the capabilities of your system allow. The only reason that you are able to determine the exact solutions of the given problem by examining the graph is that the quadratic factors over the integers and the graph crosses the -axis at marked points. Of course, this just adds to my case about this being a poor way to solve your equation since factoring the quadratic over the integers is much simpler than creating a graph of the quadratic function.

The last part of your question can be answered by following a simple rule: For any quadratic function of the form , if , then the graph is a parabola that opens upward, meaning the vertex is at the bottom and the function has a minimum. On the other hand, if , the parabola opens downward, the vertex is at the top, and the function has a maximum. In your case, , the coefficient on the high order () term is , therefore, the function has a maximum.



John