document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251930>Question 251930</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What two numbers added together equal -11 and then when the same two numbers are multiplied equal 15?\r<BR>\n" );
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document.write( "What two numbers added together equal 28 and then when those same two numbers are multiplied equal 32?\r<BR>\n" );
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document.write( "What two numbers added together equal -23 and then when those same two numbers are multiplied equal 36? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #183649 by </B><A HREF=/tutors/aboutme.mpl?userid=richwmiller><B>richwmiller(17219)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richwmiller><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=richwmiller><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=183649\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x+y=-11<BR>\n" );
document.write( "x*y=15<BR>\n" );
document.write( "factors of 15<BR>\n" );
document.write( "15,1<BR>\n" );
document.write( "5,3<BR>\n" );
document.write( "none of these add up to 11 nor have a difference of 11\r<BR>\n" );
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document.write( "x = (-11 - Sqrt[61])/2, y = (-11 + Sqrt[61])/2},<BR>\n" );
document.write( "x = (-11 + Sqrt[61])/2, y = (-11 - Sqrt[61])/2}}<BR>\n" );
document.write( "a+b=28<BR>\n" );
document.write( "a*b=32<BR>\n" );
document.write( "factors of 32<BR>\n" );
document.write( "32,1<BR>\n" );
document.write( "16,2<BR>\n" );
document.write( "8,4<BR>\n" );
document.write( "none of these add up to 28 nor have a difference of 28<BR>\n" );
document.write( "a = 2(7+sqrt(41)) and b = -2(sqrt(41)-7)<BR>\n" );
document.write( "a = -2 (-7 + Sqrt[41]), b = 2 (7 + Sqrt[41])<BR>\n" );
document.write( "a = 2 (7 + Sqrt[41]), b = -2 (-7 + Sqrt[41])<BR>\n" );
document.write( "c+d=-23<BR>\n" );
document.write( "c*d=36<BR>\n" );
document.write( "factors of 36<BR>\n" );
document.write( "36,1<BR>\n" );
document.write( "18,2<BR>\n" );
document.write( "9,4<BR>\n" );
document.write( "None of these add up to 23 nor have a difference of 23<BR>\n" );
document.write( "c = (-23 - Sqrt[385])/2, d = (-23 + Sqrt[385])/2}<BR>\n" );
document.write( "c = (-23 + Sqrt[385])/2, d = (-23 - Sqrt[385])/2}<BR>\n" );
document.write( "None of these have whole numbers as factors<BR>\n" );
document.write( "The graph is a  line crossing a hyperbola twice<BR>\n" );
document.write( "I'll go through the first one with you.\r<BR>\n" );
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document.write( "x+y=-11<BR>\n" );
document.write( "x*y=15<BR>\n" );
document.write( "The process be the same for the three problems<BR>\n" );
document.write( "y=-11-x<BR>\n" );
document.write( "x*(-11-x)=15<BR>\n" );
document.write( "-11x+x^2=15<BR>\n" );
document.write( "-x^2-11x=15<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
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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=quadratic.solver>SOLVE quadratic equation with variable</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=-1x%5E2%2B-11x%2B-15+=+0 ALIGN=MIDDLE  ALT=\"-1x%5E2%2B-11x%2B-15+=+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=%28-11%29%5E2-4%2A-1%2A-15=61 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-11%29%5E2-4%2A-1%2A-15=61\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=61 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--11%2B-sqrt%28+61+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--11%2B-sqrt%28+61+%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-%28-11%29%2Bsqrt%28+61+%29%29%2F2%5C-1+=+-9.40512483795333 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-11%29%2Bsqrt%28+61+%29%29%2F2%5C-1+=+-9.40512483795333\"   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-%28-11%29-sqrt%28+61+%29%29%2F2%5C-1+=+-1.59487516204667 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-11%29-sqrt%28+61+%29%29%2F2%5C-1+=+-1.59487516204667\"   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=-1x%5E2%2B-11x%2B-15 ALIGN=MIDDLE  ALT=\"-1x%5E2%2B-11x%2B-15\"   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=-1x%5E2%2B-11x%2B-15+=+-1%28x--9.40512483795333%29%2A%28x--1.59487516204667%29 ALIGN=MIDDLE  ALT=\"-1x%5E2%2B-11x%2B-15+=+-1%28x--9.40512483795333%29%2A%28x--1.59487516204667%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -9.40512483795333, -1.59487516204667.\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+-1%2Ax%5E2%2B-11%2Ax%2B-15+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B-11%2Ax%2B-15+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );