document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=196582>Question 196582</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi!\r<BR>\n" );
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document.write( "I'm having some difficulty with the following problem:\r<BR>\n" );
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document.write( "If F is the focal length of a convex lens and an object is placed at a distance x from the lens, then its image will be at a distance y from the lens, where F, x, and y are related by the lens equation below. Suppose that a lens has a focal length of 2.1 cm, and that the image of an object is 4 cm closer to the lens than the object itself. How far from the lens is the object?\r<BR>\n" );
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document.write( "1/F = 1/x + 1/y \r<BR>\n" );
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document.write( "thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #147331 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=147331\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Since \"the image of an object is 4 cm closer to the lens than the object itself\", this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x-4 ALIGN=MIDDLE  ALT=\"y=x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2FF+=+1%2Fx+%2B+1%2Fy+ ALIGN=MIDDLE  ALT=\"1%2FF+=+1%2Fx+%2B+1%2Fy+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2.1+=+1%2Fx+%2B+1%2F%28x-4%29 ALIGN=MIDDLE  ALT=\"1%2F2.1+=+1%2Fx+%2B+1%2F%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=2.1 ALIGN=MIDDLE  ALT=\"F=2.1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x-4 ALIGN=MIDDLE  ALT=\"y=x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%282.1%29x%28x-4%29%281%2Fcross%282.1%29%29+=+2.1%2Across%28x%29%28x-4%29%281%2Fcross%28x%29%29+%2B+2.1x%2Across%28%28x-4%29%29%281%2Fcross%28%28x-4%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply EVERY term by the LCD <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.1x%28x-4%29 ALIGN=MIDDLE  ALT=\"2.1x%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to clear out the fractions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x-4%29=2.1%28x-4%29%2B2.1x%281%29 ALIGN=MIDDLE  ALT=\"x%28x-4%29=2.1%28x-4%29%2B2.1x%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel out and simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x-4%29=2.1%28x-4%29%2B2.1x ALIGN=MIDDLE  ALT=\"x%28x-4%29=2.1%28x-4%29%2B2.1x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4x=2.1x-8.4%2B2.1x ALIGN=MIDDLE  ALT=\"x%5E2-4x=2.1x-8.4%2B2.1x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%5E2-40x=21x-84%2B21x ALIGN=MIDDLE  ALT=\"10x%5E2-40x=21x-84%2B21x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply EVERY term by 10 to make every number a whole number.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%5E2-40x-21x%2B84-21x=0 ALIGN=MIDDLE  ALT=\"10x%5E2-40x-21x%2B84-21x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Get all terms to the left side.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%5E2-82x%2B84=0 ALIGN=MIDDLE  ALT=\"10x%5E2-82x%2B84=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "Notice we have a quadratic in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax%5E2%2BBx%2BC ALIGN=MIDDLE  ALT=\"Ax%5E2%2BBx%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10 ALIGN=MIDDLE  ALT=\"A=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-82 ALIGN=MIDDLE  ALT=\"B=-82\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=84 ALIGN=MIDDLE  ALT=\"C=84\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for x\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 ALIGN=MIDDLE  ALT=\"x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-82%29+%2B-+sqrt%28+%28-82%29%5E2-4%2810%29%2884%29+%29%29%2F%282%2810%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-82%29+%2B-+sqrt%28+%28-82%29%5E2-4%2810%29%2884%29+%29%29%2F%282%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=10 ALIGN=MIDDLE  ALT=\"A=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-82 ALIGN=MIDDLE  ALT=\"B=-82\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=84 ALIGN=MIDDLE  ALT=\"C=84\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2882+%2B-+sqrt%28+%28-82%29%5E2-4%2810%29%2884%29+%29%29%2F%282%2810%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2882+%2B-+sqrt%28+%28-82%29%5E2-4%2810%29%2884%29+%29%29%2F%282%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-82 ALIGN=MIDDLE  ALT=\"-82\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=82 ALIGN=MIDDLE  ALT=\"82\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2882+%2B-+sqrt%28+6724-4%2810%29%2884%29+%29%29%2F%282%2810%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2882+%2B-+sqrt%28+6724-4%2810%29%2884%29+%29%29%2F%282%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-82 ALIGN=MIDDLE  ALT=\"-82\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6724 ALIGN=MIDDLE  ALT=\"6724\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2882+%2B-+sqrt%28+6724-3360+%29%29%2F%282%2810%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2882+%2B-+sqrt%28+6724-3360+%29%29%2F%282%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2810%29%2884%29 ALIGN=MIDDLE  ALT=\"4%2810%29%2884%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3360 ALIGN=MIDDLE  ALT=\"3360\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2882+%2B-+sqrt%28+3364+%29%29%2F%282%2810%29%29 ALIGN=MIDDLE  ALT=\"x+=+%2882+%2B-+sqrt%28+3364+%29%29%2F%282%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3360 ALIGN=MIDDLE  ALT=\"3360\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6724 ALIGN=MIDDLE  ALT=\"6724\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3364 ALIGN=MIDDLE  ALT=\"3364\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2882+%2B-+sqrt%28+3364+%29%29%2F%2820%29 ALIGN=MIDDLE  ALT=\"x+=+%2882+%2B-+sqrt%28+3364+%29%29%2F%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2882+%2B-+58%29%2F%2820%29 ALIGN=MIDDLE  ALT=\"x+=+%2882+%2B-+58%29%2F%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3364 ALIGN=MIDDLE  ALT=\"3364\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=58 ALIGN=MIDDLE  ALT=\"58\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2882+%2B+58%29%2F%2820%29 ALIGN=MIDDLE  ALT=\"x+=+%2882+%2B+58%29%2F%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2882+-+58%29%2F%2820%29 ALIGN=MIDDLE  ALT=\"x+=+%2882+-+58%29%2F%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28140%29%2F%2820%29 ALIGN=MIDDLE  ALT=\"x+=+%28140%29%2F%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=++%2824%29%2F%2820%29 ALIGN=MIDDLE  ALT=\"x+=++%2824%29%2F%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+7 ALIGN=MIDDLE  ALT=\"x+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+6%2F5 ALIGN=MIDDLE  ALT=\"x+=+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. \r<BR>\n" );
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document.write( "So the <i>possible</i> answers are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+7 ALIGN=MIDDLE  ALT=\"x+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+6%2F5 ALIGN=MIDDLE  ALT=\"x+=+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
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document.write( "  \r<BR>\n" );
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document.write( "However, since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F5-4=1.2-4=-2.8 ALIGN=MIDDLE  ALT=\"6%2F5-4=1.2-4=-2.8\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which doesn't make much sense), this means we'll ignore  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+6%2F5 ALIGN=MIDDLE  ALT=\"x+=+6%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the only answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+7 ALIGN=MIDDLE  ALT=\"x+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the object is 7 cm from the lens. </SPAN>\n" );
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