document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=627671>Question 627671</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a woman is 4 times older than her daughter, 6 years ago the product of their ages was 136. find their present age </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #395057 by </B><A HREF=/tutors/aboutme.mpl?userid=Maths68><B>Maths68(1474)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Maths68><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=Maths68><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=395057\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let<BR>\n" );
document.write( "Present age of woman = w years old<BR>\n" );
document.write( "Present age of daughter = d years old\r<BR>\n" );
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document.write( "woman is 4 times older than her daughter<BR>\n" );
document.write( "w=4d............(1)<BR>\n" );
document.write( "6 years ago <BR>\n" );
document.write( "woman was = w-6 years old<BR>\n" );
document.write( "daughter was = d-6<BR>\n" );
document.write( "then<BR>\n" );
document.write( "product of their ages was 136<BR>\n" );
document.write( "(w-6)(d-6)=136<BR>\n" );
document.write( "substitute the value of from (1) to above equation<BR>\n" );
document.write( "(4d-6)(d-6)=136<BR>\n" );
document.write( "4d^2-24d-6d+36=136<BR>\n" );
document.write( "4d^2-30d+36=136<BR>\n" );
document.write( "2(2d^2-15d+18)=136<BR>\n" );
document.write( "Divide by 2 both sides<BR>\n" );
document.write( "2(2d^2-15d+18)/2=136/2<BR>\n" );
document.write( "2d^2-15d+18=68<BR>\n" );
document.write( "2d^2-15d+18-68=0<BR>\n" );
document.write( "2d^2-15d-50=0<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=ad%5E2%2Bbd%2Bc=0 ALIGN=MIDDLE  ALT=\"ad%5E2%2Bbd%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2d%5E2%2B-15d%2B-50+=+0 ALIGN=MIDDLE  ALT=\"2d%5E2%2B-15d%2B-50+=+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=d%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"d%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=%28-15%29%5E2-4%2A2%2A-50=625 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-15%29%5E2-4%2A2%2A-50=625\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=625 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--15%2B-sqrt%28+625+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--15%2B-sqrt%28+625+%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=d%5B1%5D+=+%28-%28-15%29%2Bsqrt%28+625+%29%29%2F2%5C2+=+10 ALIGN=MIDDLE  ALT=\"d%5B1%5D+=+%28-%28-15%29%2Bsqrt%28+625+%29%29%2F2%5C2+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5B2%5D+=+%28-%28-15%29-sqrt%28+625+%29%29%2F2%5C2+=+-2.5 ALIGN=MIDDLE  ALT=\"d%5B2%5D+=+%28-%28-15%29-sqrt%28+625+%29%29%2F2%5C2+=+-2.5\"   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=2d%5E2%2B-15d%2B-50 ALIGN=MIDDLE  ALT=\"2d%5E2%2B-15d%2B-50\"   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=2d%5E2%2B-15d%2B-50+=+2%28d-10%29%2A%28d--2.5%29 ALIGN=MIDDLE  ALT=\"2d%5E2%2B-15d%2B-50+=+2%28d-10%29%2A%28d--2.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 10, -2.5.\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+2%2Ax%5E2%2B-15%2Ax%2B-50+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-15%2Ax%2B-50+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "d=10 or d=-2.5 (unacceptable)<BR>\n" );
document.write( "so<BR>\n" );
document.write( "d=10<BR>\n" );
document.write( "Put the value of d in (1)<BR>\n" );
document.write( "w=4d<BR>\n" );
document.write( "w=4(10)<BR>\n" );
document.write( "w=40\r<BR>\n" );
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document.write( "Present age of woman = w = 40 years old<BR>\n" );
document.write( "Present age of daughter = d = 10 years old<BR>\n" );
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