document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=324557>Question 324557</A>: <I><SPAN STYLE=\"word-break: break-all;\"> factor completely (x+3)^2-10(x+3)+24\r<BR>\n" );
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document.write( "write a quadratic equation and answer the following: the shortest leg of a right triangle is two less than the hypotenuse. the hypotenuse is one longer than the longest leg.\r<BR>\n" );
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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 #232292 by </B><A HREF=/tutors/aboutme.mpl?userid=mathslover><B>mathslover(157)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathslover><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=mathslover><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=232292\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (x+3)^2-10(x+3)+24\r<BR>\n" );
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document.write( "Let's replace x+3 by A <BR>\n" );
document.write( "So, the given expression would transform to \r<BR>\n" );
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document.write( "A^2 -10A + 24<BR>\n" );
document.write( "Rewriting it as\r<BR>\n" );
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document.write( "A^2 - 6A -4A + 24<BR>\n" );
document.write( "A(A-6) -4(A-6)<BR>\n" );
document.write( "Taking A-6 as the common factor,<BR>\n" );
document.write( "(A-6)(A-4)<BR>\n" );
document.write( "Substituting A=x+3 back into the expression we get\r<BR>\n" );
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document.write( "(x+3-6)(x+3-4)<BR>\n" );
document.write( "(x-3)(x-1) is the required factor.<BR>\n" );
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document.write( "write a quadratic equation and answer the following: the shortest leg of a right triangle is two less than the hypotenuse. the hypotenuse is one longer than the longest leg.\r<BR>\n" );
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document.write( "Let the longest leg be x<BR>\n" );
document.write( "So the hypotenuse is (x+1) {the hypotenuse is one longer than the longest leg}<BR>\n" );
document.write( "And the shortest leg is (x+1)-2=x-1 {the shortest leg of a right triangle is two<BR>\n" );
document.write( "                               less than the hypotenuse}<BR>\n" );
document.write( "Using Pythagoras's theorem,<BR>\n" );
document.write( "hypotenuse^2 = shorter leg^2 + longer leg^2<BR>\n" );
document.write( "we get<BR>\n" );
document.write( "(x+1)^2 = (x-1)^2 + x^2<BR>\n" );
document.write( "Expanding the LHS and the RHS <BR>\n" );
document.write( "x^2 + 2x+ 1 = x^2 -2x+1 + x^2<BR>\n" );
document.write( "Moving the Right hand side expression to Left hand side we get<BR>\n" );
document.write( "x^2 + 2x+ 1 - (x^2 -2x+1 + x^2)= 0<BR>\n" );
document.write( "grouping the like terms,<BR>\n" );
document.write( "(x^2 - x^2 -x^2) + (2x + 2x ) + (1 -1)=0<BR>\n" );
document.write( "-x^2 + 4x =0<BR>\n" );
document.write( "-x(x-4)=0\r<BR>\n" );
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document.write( "which gives x=0 and x=4 as two possible solutions. But we cannot have a triangle with side of length 0. So discarding x=0 we get the other solution as x=4<BR>\n" );
document.write( "Hence, the sides of the triangle are<BR>\n" );
document.write( "The longest side=4<BR>\n" );
document.write( "The shorter side=4-1=3<BR>\n" );
document.write( "hypotenuse=4+1=5\r<BR>\n" );
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document.write( " </SPAN>\n" );
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