document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=801179>Question 801179</A>: <I><SPAN STYLE=\"word-break: break-all;\"> two cars leave an intersection, one traveling west and the other south. After some time the slower car is 7 mi nearer to the intersection than the faster car. At that time, two cars are 17 mi apart. How far did each car travel? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #483483 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=483483\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> West and South are directions perpendicular to each other.<BR>\n" );
document.write( "We assume that the trajectories of the two cars are perpendicular straight lines.<BR>\n" );
document.write( "(Those must be long, straight North-South and East-West roads, like the ones I have seen in South Dakota).<BR>\n" );
document.write( "We can call one car the slower car, and the other car the faster car, but there is no way to know which one was going West and which one was going South.\r<BR>\n" );
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document.write( "At a certain time after both cars have left the intersection, the distances traveled after leaving the intersection, and the distance between the two car, all measured in miles, are as follows:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= distance traveled by the faster car<BR>\n" );
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;\" >= distance traveled by the slower car<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17 ALIGN=MIDDLE  ALT=\"17\"   DISABLED_style= \"max-width:90%; height:auto;\" >= distance between the two cars<BR>\n" );
document.write( "It's either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-16%2C2%2C-10%2C8%2C%0D%0Aarrow%28-8%2C2%2C-8%2C5%29%2C+locate%28-8.2%2C5.8%2CN%29%2C%0D%0Aline%28-16%2C0%2C2%2C0%29%2Cline%280%2C-10%2C0%2C8%29%2Crectangle%28-0.5%2C0%2C0%2C-0.5%29%2C%0D%0Ared%28circle%28-15%2C0%2C0.3%29%29%2Cgreen%28circle%280%2C-8%2C0.3%29%29%2C%0D%0Alocate%28-15.5%2C1.1%2Ccar%29%2Clocate%28-1.8%2C-7.6%2Ccar%29%2C%0D%0Aline%28-15%2C0%2C0%2C-8%29%2Clocate%28-8%2C-3%2C17%29%2C%0D%0Alocate%28-7.5%2C0%2Cx%29%2Clocate%28-1.8%2C-3.6%2Cx-7%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-9%2C9%2C-16%2C2%2C%0D%0Aarrow%285%2C-8%2C5%2C-5%29%2C+locate%284.8%2C-4.1%2CN%29%2C%0D%0Aline%28-9%2C0%2C9%2C0%29%2Cline%280%2C-16%2C0%2C2%29%2Crectangle%28-0.5%2C0%2C0%2C-0.5%29%2C%0D%0Ared%28circle%280%2C-15%2C0.3%29%29%2Cgreen%28circle%28-8%2C0%2C0.3%29%29%2C%0D%0Alocate%28-8.5%2C1.1%2Ccar%29%2Clocate%28-1.8%2C-14.6%2Ccar%29%2C%0D%0Aline%28-8%2C0%2C0%2C-15%29%2Clocate%28-4.5%2C-6%2C17%29%2C%0D%0Alocate%28-4.5%2C0%2Cx-7%29%2Clocate%28-0.6%2C-6%2Cx%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Either way, we have a right triangle with leg lengths <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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;\" > and hypotenuse <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17 ALIGN=MIDDLE  ALT=\"17\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "According to the Pythagorean theorem,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%28x-7%29%5E2=17%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B%28x-7%29%5E2=17%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solving that equation for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%28x-7%29%5E2=17%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B%28x-7%29%5E2=17%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx%5E2-14x%2B49=289 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx%5E2-14x%2B49=289\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-14x%2B49-289=0 ALIGN=MIDDLE  ALT=\"2x%5E2-14x%2B49-289=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-14x-240=0 ALIGN=MIDDLE  ALT=\"2x%5E2-14x-240=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing both sides of the equation by 2, we simplify to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-7x-120=0 ALIGN=MIDDLE  ALT=\"x%5E2-7x-120=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Factoring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-7x-120=%28x-15%29%28x%2B8%29 ALIGN=MIDDLE  ALT=\"x%5E2-7x-120=%28x-15%29%28x%2B8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we transform the equation into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-15%29%28x%2B8%29=0 ALIGN=MIDDLE  ALT=\"%28x-15%29%28x%2B8%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve it by factoring.<BR>\n" );
document.write( "The solutions for that equation are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-8 ALIGN=MIDDLE  ALT=\"x=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > derived from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B8=0 ALIGN=MIDDLE  ALT=\"x%2B8=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which does not make sense, because we expect a positive distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=15%29 ALIGN=MIDDLE  ALT=\"highlight%28x=15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-15=0 ALIGN=MIDDLE  ALT=\"x-15=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=15 ALIGN=MIDDLE  ALT=\"x=15\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-7=15-7 ALIGN=MIDDLE  ALT=\"x-7=15-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x-7=8%29 ALIGN=MIDDLE  ALT=\"highlight%28x-7=8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The cars have traveled <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%288%29 ALIGN=MIDDLE  ALT=\"highlight%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2815%29 ALIGN=MIDDLE  ALT=\"highlight%2815%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles away from the intersection.<BR>\n" );
document.write( " <BR>\n" );
document.write( "OTHER OPTIONS:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-7x-120=0 ALIGN=MIDDLE  ALT=\"x%5E2-7x-120=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > can also be solved by applying the quadratic formula, or by \"completing the square.\" For this particular equation, the quadratic formula may be more popular.<BR>\n" );
document.write( "The quadratic formula says that the solutions to an equation of the form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are given by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-7x-120=0 ALIGN=MIDDLE  ALT=\"x%5E2-7x-120=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-7 ALIGN=MIDDLE  ALT=\"b=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-120 ALIGN=MIDDLE  ALT=\"c=-120\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-7%29+%2B-+sqrt%28%28-7%29%5E2-4%2A1%2A%28-120%29+%29%29%2F%282%2A1%29=%287+%2B-+sqrt%2849%2B480%29%29%2F2=%287+%2B-+sqrt%28529%29%29%2F2=%287+%2B-+23%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the negative solution <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%287-23%29%2F2 ALIGN=MIDDLE  ALT=\"x=%287-23%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not make sense as a distance,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%287%2B23%29%2F2 ALIGN=MIDDLE  ALT=\"x=%287%2B23%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=30%2F2 ALIGN=MIDDLE  ALT=\"x=30%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=15%29 ALIGN=MIDDLE  ALT=\"highlight%28x=15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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