document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=107036>Question 107036</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi,\r<BR>\n" );
document.write( "\n" );
document.write( "Please help me solve this problem:\r<BR>\n" );
document.write( "\n" );
document.write( "The formula for computing the amount A of an investment of principal P invested at interest rate r for 1 year and compounded semiannually is A=P(1+r/2)^2.  Approximately what interest rate is necessary for $1,000 to grow to $1,075 in 1 year if the interest is compounded semiannually?\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #77891 by </B><A HREF=/tutors/aboutme.mpl?userid=edjones><B>edjones(8007)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=edjones><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=edjones><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=77891\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A=P(1+r/2)^2<BR>\n" );
document.write( "1072=1000(.5r+1)^2<BR>\n" );
document.write( "1072=1000(.25r^2+r+1)<BR>\n" );
document.write( "1072=250r^2+1000r+1000<BR>\n" );
document.write( "250r^2+1000r-72=0<BR>\n" );
document.write( "2(125r^2+500r-36)=0<BR>\n" );
document.write( "r=-4.07075 not an answer because the rate has to be positive.<BR>\n" );
document.write( "r=.07075...(see below)<BR>\n" );
document.write( "Check:<BR>\n" );
document.write( "1000(1+.070749/2)^2=1000*(1.035374...)^2=1000*1.072=1072<BR>\n" );
document.write( "Ed<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <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;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125x%5E2%2B500x%2B-36+=+0 ALIGN=MIDDLE  ALT=\"125x%5E2%2B500x%2B-36+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28500%29%5E2-4%2A125%2A-36=268000 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28500%29%5E2-4%2A125%2A-36=268000\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=268000 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-500%2B-sqrt%28+268000+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-500%2B-sqrt%28+268000+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28500%29%2Bsqrt%28+268000+%29%29%2F2%5C125+=+0.0707486568871655 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28500%29%2Bsqrt%28+268000+%29%29%2F2%5C125+=+0.0707486568871655\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28500%29-sqrt%28+268000+%29%29%2F2%5C125+=+-4.07074865688717 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28500%29-sqrt%28+268000+%29%29%2F2%5C125+=+-4.07074865688717\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125x%5E2%2B500x%2B-36 ALIGN=MIDDLE  ALT=\"125x%5E2%2B500x%2B-36\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125x%5E2%2B500x%2B-36+=+%28x-0.0707486568871655%29%2A%28x--4.07074865688717%29 ALIGN=MIDDLE  ALT=\"125x%5E2%2B500x%2B-36+=+%28x-0.0707486568871655%29%2A%28x--4.07074865688717%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.0707486568871655, -4.07074865688717.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+125%2Ax%5E2%2B500%2Ax%2B-36+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+125%2Ax%5E2%2B500%2Ax%2B-36+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );