document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=475466>Question 475466</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you please help me solve this word problem involving quadratic equations, I am lost and can't figure out where to start. \r<BR>\n" );
document.write( "\n" );
document.write( "The function D(x) models the cumulative number of deaths from a disease x years after 1984, estimate the year when there were 100,000 deaths. \r<BR>\n" );
document.write( "\n" );
document.write( "D(x) =2285x^2 + 5583x + 5540. \r<BR>\n" );
document.write( "\n" );
document.write( "There were approximatley 100,000 in the year __________________<BR>\n" );
document.write( "                                       (round to the nearest year) \r<BR>\n" );
document.write( "\n" );
document.write( "Your help will be appreciated!<BR>\n" );
document.write( "Thanks Abby </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #326067 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=326067\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the equation is D(x) = 2285x^2 + 5583x + 5540<BR>\n" );
document.write( "if you graph the equation, just replace D(x) with y and you have the equation that is ready for graphing.<BR>\n" );
document.write( "that equation is y = 2285x^2 + 5583x + 5540<BR>\n" );
document.write( "it looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-2%2C13%2C-10000%2C150000%2C2285x%5E2+%2B+5583x+%2B+5540%2C100000%2C5540%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "from the graph, it looks like the cumulative number of deaths will reach 100,000 somewhere around the 5th year.<BR>\n" );
document.write( "the formula to find that would be the formula you were given, but replacing D(x) with 100,000l, because, when the cumulative deaths reach 100,000, this means that D(x) = 100,000<BR>\n" );
document.write( "The equation becomes:<BR>\n" );
document.write( "100,000 = 2285x^2 + 5583x + 5540<BR>\n" );
document.write( "subtract 100,000 from both sides of this equation to get:<BR>\n" );
document.write( "2285x^2 + 5583x + 5540 - 100,000 = 0<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "2285x^2 + 5583x - 94460 = 0<BR>\n" );
document.write( "this is a quadratic equation that needs to be solved to give you your answer.<BR>\n" );
document.write( "solve using the quadratic formula.<BR>\n" );
document.write( "that formula is:<BR>\n" );
document.write( "x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29%29 ALIGN=MIDDLE  ALT=\"%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a = 2285<BR>\n" );
document.write( "b = 5583<BR>\n" );
document.write( "c = -94460<BR>\n" );
document.write( "-b = -5583<BR>\n" );
document.write( "b^2 = 31169889<BR>\n" );
document.write( "4ac = -863364400<BR>\n" );
document.write( "b^2 - 4ac = 894534289<BR>\n" );
document.write( "sqrt(b^2-4ac) = 29908.76609<BR>\n" );
document.write( "2a = 4570<BR>\n" );
document.write( "x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29%29 ALIGN=MIDDLE  ALT=\"%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes:<BR>\n" );
document.write( "x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-5583%2B-29908.76609%29%2F%284570%29%29 ALIGN=MIDDLE  ALT=\"%28-5583%2B-29908.76609%29%2F%284570%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "this gets you:<BR>\n" );
document.write( "x = 5.322924746 <BR>\n" );
document.write( "or:<BR>\n" );
document.write( "x = -7.766250785<BR>\n" );
document.write( "Since x can only be positive, this means that x can only be equal to 5.322924746.<BR>\n" );
document.write( "replacing x in the equation with 5.322924746 gets you:<BR>\n" );
document.write( "D(x) = 2285x^2 + 5583x + 5540 becomes:<BR>\n" );
document.write( "D(5.322924746) = 100,000<BR>\n" );
document.write( "the graph and the equations check out ok so that's your answer.<BR>\n" );
document.write( "The cumulative number of deaths from a disease will reach 100,000 5.322924746 years after 1984.<BR>\n" );
document.write( "The fact that you are dealing with a quadratic equation might have thrown you off.<BR>\n" );
document.write( "That starts off high and drops to a minimum value and then ends up high.<BR>\n" );
document.write( "If we just extend the graph a little more to the left, you will see that better.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-13%2C13%2C-10000%2C150000%2C2285x%5E2+%2B+5583x+%2B+5540%2C100000%2C5540%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "While that is a graph of a quadratic equation, and the equation starts off high and then drops to a minimum value and then goes up again, the only values of y in that equation are when x is greater than 0.<BR>\n" );
document.write( "Look at the graph and you will see that the graph is rising from x = 0 and up.<BR>\n" );
document.write( "Furthermore, you should see that y = 5540 when x is equal to 0.<BR>\n" );
document.write( "That is the number of deaths that occurred in 1984.<BR>\n" );
document.write( "x = 0 represents 1984.<BR>\n" );
document.write( "x = 1 represents 1985.<BR>\n" );
document.write( "etc.<BR>\n" );
document.write( "You needed the quadratic formula to solve this problem.<BR>\n" );
document.write( "With those large numbers, it was a little clumsy, but it worked fine.<BR>\n" );
document.write( "you needed to start off with setting the equation equal to 100,000 as we did early on.<BR>\n" );
document.write( "you need to then transform that equation into the standard form of a quadratic equation which we did by subtracting 100,000 from both sides of the equation.<BR>\n" );
document.write( "once you got it into the standard form, it was simply the application of the quadratic formula to get the roots.<BR>\n" );
document.write( "Here's a good reference on solving quadratic equations to find the roots.<BR>\n" );
document.write( "It discusses the 3 main methods:<BR>\n" );
document.write( "The quadratic formula method is near the bottom.<BR>\n" );
document.write( "It's all good information, but just pick out what you need for now.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut17_quad.htm\" target = \"_blank\">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut17_quad.htm</a> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );