Question 98320: Solve 3x(squared)-8x+4=0 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.
a. 2, between 0 and 1
b. between 0 and 1; between 7 and 8
c. 1,2
c. between 0 and 1, between 3 and 4.
Thank you for helping this old grandmother out.
Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website! The two roots are x = 2 and x = 2/3 so answer "a" is correct.
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But more important, how do you solve this since you are supposed to do it by graphing.
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You start with the equation and you plug in values of x that are
of your choosing. Then you plot the (x,y) points that result. The points will give you
an idea of what the graph looks like, and once you get the general idea, you will be able
to narrow down the values of x you need to try to get close to the answer.
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Let's do a few points to give you the idea.
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We'll begin by letting x equal zero, because that's an easy one to do. When x is equal to
zero, all the terms that have an x in them go to zero so you are left with:
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That means that the (x,y) point (0,4) is on the graph. Plot it.
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Next, let's try letting x = +1 because the resulting values are relatively easy to calculate.
The equation becomes:
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This means that when x = 1, y = -1. So the point (1, -1) is on the graph. Plot that point
on the same graph that you did the point (0, 4). You now you can notice a couple of significant
things from just these two points. The first significant thing is that the graph has gone down
from the first point (0, 4) to the second point (1, -1). In fact if you draw a line between
those two points, you can see that the line has to cross the x-axis. And roots occur at
values of x where the graph crosses the x-axis. From your line connecting the first two
points of your graph, you can tell that the graph is going to cross the x-axis at somewhere
between x = 0 and x = 1.
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Next, let's try letting x equal 3 and see what we get as the corresponding value of y. The
equation becomes:
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This tells us that when x = 3, the corresponding value of y is +7. So the (x,y) point (3, 7)
is on the graph. Plot it and note something significant. It is back above the x-axis. So
to get from the point (1, -1) that you plotted as your second point to this point (3, 7)
the graph is going to cross the x-axis again and at that crossing point you have the second
root. You might want to try letting x = 2 to get a little better idea of the location of
the crossing point. If you do, you will find that when x equals 2 the corresponding
value of y is 0. This means that the graph crosses the x-axis exactly at that point ...
meaning that x = 2 is a root of the equation.
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And by graphing you may want to try a few values of x between x = 0 and x = 1 to get a better
idea of where the crossing point is in that space. Remember, what you are trying to
do is to get the value of y as close to zero as possible because the closer it is to zero,
the closer you are to having the correct value of x for the root.
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As a final thing, to help you visualize what you are doing, here is the graph of the given
equation . You can use it to do a quick check on the (x,y) points that
you calculate.
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Hope this helps you to understand the problem a little better.
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