document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=98320>Question 98320</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve 3x(squared)-8x+4=0 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.\r<BR>\n" );
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document.write( "a. 2, between 0 and 1<BR>\n" );
document.write( "b. between 0 and 1; between 7 and 8<BR>\n" );
document.write( "c. 1,2<BR>\n" );
document.write( "c. between 0 and 1, between 3 and 4.\r<BR>\n" );
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document.write( "Thank you for helping this old grandmother out. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #71509 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=71509\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The two roots are x = 2 and x = 2/3 so answer \"a\" is correct.<BR>\n" );
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document.write( "But more important, how do you solve this since you are supposed to do it by graphing.<BR>\n" );
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document.write( "You start with the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3x%5E2+-8x+%2B+4 ALIGN=MIDDLE  ALT=\"y+=+3x%5E2+-8x+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you plug in values of x that are<BR>\n" );
document.write( "of your choosing.  Then you plot the (x,y) points that result. The points will give you<BR>\n" );
document.write( "an idea of what the graph looks like, and once you get the general idea, you will be able<BR>\n" );
document.write( "to narrow down the values of x you need to try to get close to the answer.<BR>\n" );
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document.write( "Let's do a few points to give you the idea.<BR>\n" );
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document.write( "We'll begin by letting x equal zero, because that's an easy one to do. When x is equal to<BR>\n" );
document.write( "zero, all the terms that have an x in them go to zero so you are left with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3%280%29%5E2+-+8%2A%280%29+%2B+4+=+4 ALIGN=MIDDLE  ALT=\"y+=+3%280%29%5E2+-+8%2A%280%29+%2B+4+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "That means that the (x,y) point (0,4) is on the graph. Plot it.<BR>\n" );
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document.write( "Next, let's try letting x = +1 because the resulting values are relatively easy to calculate. <BR>\n" );
document.write( "The equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3%281%29%5E2+-+8%281%29+%2B+4+=+3+-+8+%2B+4+=+-1 ALIGN=MIDDLE  ALT=\"y+=+3%281%29%5E2+-+8%281%29+%2B+4+=+3+-+8+%2B+4+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This means that when x = 1, y = -1. So the point (1, -1) is on the graph. Plot that point <BR>\n" );
document.write( "on the same graph that you did the point (0, 4). You now you can notice a couple of significant<BR>\n" );
document.write( "things from just these two points. The first significant thing is that the graph has gone down<BR>\n" );
document.write( "from the first point (0, 4) to the second point (1, -1). In fact if you draw a line between<BR>\n" );
document.write( "those two points, you can see that the line has to cross the x-axis. And roots occur at<BR>\n" );
document.write( "values of x where the graph crosses the x-axis. From your line connecting the first two<BR>\n" );
document.write( "points of your graph, you can tell that the graph is going to cross the x-axis at somewhere<BR>\n" );
document.write( "between x = 0 and x = 1.  <BR>\n" );
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document.write( "Next, let's try letting x equal 3 and see what we get as the corresponding value of y. The<BR>\n" );
document.write( "equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3%283%29%5E2-8%2A%283%29+%2B+4+=+27+-+24+%2B+4+=+%2B7 ALIGN=MIDDLE  ALT=\"y+=+3%283%29%5E2-8%2A%283%29+%2B+4+=+27+-+24+%2B+4+=+%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This tells us that when x = 3, the corresponding value of y is +7. So the (x,y) point (3, 7) <BR>\n" );
document.write( "is on the graph. Plot it and note something significant. It is back above the x-axis. So<BR>\n" );
document.write( "to get from the point (1, -1) that you plotted as your second point to this point (3, 7)<BR>\n" );
document.write( "the graph is going to cross the x-axis again and at that crossing point you have the second<BR>\n" );
document.write( "root. You might want to try letting x = 2 to get a little better idea of the location of<BR>\n" );
document.write( "the crossing point. If you do, you will find that when x equals 2 the corresponding<BR>\n" );
document.write( "value of y is 0. This means that the graph crosses the x-axis exactly at that point ...<BR>\n" );
document.write( "meaning that x = 2 is a root of the equation.<BR>\n" );
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document.write( "And by graphing you may want to try a few values of x between x = 0 and x = 1 to get a better<BR>\n" );
document.write( "idea of where the crossing point is in that space.  Remember, what you are trying to<BR>\n" );
document.write( "do is to get the value of y as close to zero as possible because the closer it is to zero,<BR>\n" );
document.write( "the closer you are to having the correct value of x for the root.<BR>\n" );
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document.write( "As a final thing, to help you visualize what you are doing, here is the graph of the given<BR>\n" );
document.write( "equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3x%5E2+-8x%2B4 ALIGN=MIDDLE  ALT=\"y+=+3x%5E2+-8x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. You can use it to do a quick check on the (x,y) points that<BR>\n" );
document.write( "you calculate.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28600%2C+600%2C+-4%2C+4%2C+-5%2C+8%2C+3x%5E2-8x%2B4%29 ALIGN=MIDDLE  ALT=\"graph+%28600%2C+600%2C+-4%2C+4%2C+-5%2C+8%2C+3x%5E2-8x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Hope this helps you to understand the problem a little better.<BR>\n" );
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document.write( " </SPAN>\n" );
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