document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=77881>Question 77881</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Find the length of a shorter leg of a right triangle if the longer leg is 10 miles more than the shorter leg and the hypotemuse is 10 miles less than twice the shorter leg.\r<BR>\n" );
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document.write( "2. The sum of two numbers is twentyfive and the sum of their square is three hundred and five.  Find the number. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #55990 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><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=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=55990\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. Find the length of a shorter leg of a right triangle 1. Find the length of a shorter leg of a right triangle if the longer leg is 10 miles more than the shorter leg and the hypotenuse is 10 miles less than twice the shorter leg.<BR>\n" );
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document.write( "Using pythag: a^2 + b^2 = c^2<BR>\n" );
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document.write( "Let x = a: the shorter leg<BR>\n" );
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document.write( "\"longer leg is 10 miles more than the shorter leg\"<BR>\n" );
document.write( "b = (x+10)<BR>\n" );
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document.write( "\"the hypotenuse is 10 miles less than twice the shorter leg.\"<BR>\n" );
document.write( "c = (2x - 10)<BR>\n" );
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document.write( "a^2 + b^2 = c^2 is:<BR>\n" );
document.write( "x^2 + (x+10)^2 = (2x-10)^2<BR>\n" );
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document.write( "x^2 + (x^2 + 20x + 100) = 4x^2 - 40x + 100<BR>\n" );
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document.write( "2x^2 + 20x + 100 = 4x^2 - 40x + 100<BR>\n" );
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document.write( "0 = 4x^2 - 2x^2 - 40x - 20x + 100 - 100<BR>\n" );
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document.write( "2x^2 - 60x = 0; a simple equation<BR>\n" );
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document.write( "Factors to:<BR>\n" );
document.write( "2x(x - 30) = 0<BR>\n" );
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document.write( "2x = 0<BR>\n" );
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document.write( "x = 30 the length of the shorter side<BR>\n" );
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document.write( "Check; a^2 + b^2 = c^2:<BR>\n" );
document.write( "30^2 + 40^2 = (60-10)^2<BR>\n" );
document.write( "900 + 1600 = 50^2 = 2500<BR>\n" );
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document.write( "2. \"The sum of two numbers is twenty five\"<BR>\n" );
document.write( "x + y = 25<BR>\n" );
document.write( "y = (25 - x)<BR>\n" );
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document.write( "\"the sum of their squares is three hundred and five\"<BR>\n" );
document.write( "x^2 + y^2 = 305<BR>\n" );
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document.write( "Find the number.<BR>\n" );
document.write( "Substitute (25-x) for y in the 2nd equation:<BR>\n" );
document.write( "x^2 + (25-x)^2 = 305<BR>\n" );
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document.write( "x^2 + 625 - 50x + x^2 = 305<BR>\n" );
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document.write( "2x^2 - 50x + 625 - 305 = 0<BR>\n" );
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document.write( "2x^2 - 50x + 320 = 0<BR>\n" );
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document.write( "There is no solution: Look at the discriminant of this equation:<BR>\n" );
document.write( " -b^2 - 4 * a * c<BR>\n" );
document.write( "-50^2 - 4 * 2 * 320<BR>\n" );
document.write( "2500 - 2560 = -60; <0 means x has no real solutions<BR>\n" );
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document.write( "If we graph the two equations:<BR>\n" );
document.write( "y = 25-x<BR>\n" );
document.write( "y = Sqrt(305 - x^2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-2%2C+20%2C+-2%2C+20%2C+25-x%2C+sqrt%28305-x%5E2%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-2%2C+20%2C+-2%2C+20%2C+25-x%2C+sqrt%28305-x%5E2%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "No point of intersection<BR>\n" );
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