document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=705705>Question 705705</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the roots of 2x^2 +3x-1=0 are alpha and beta. Find the values of alpha square + beta square </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #434730 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=434730\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+%2B3x-1=0 ALIGN=MIDDLE  ALT=\"2x%5E2+%2B3x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%2Bbeta=-3%2F2 ALIGN=MIDDLE  ALT=\"alpha%2Bbeta=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28alpha%29%28beta%29=-1%2F2 ALIGN=MIDDLE  ALT=\"%28alpha%29%28beta%29=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(Reasons below)<BR>\n" );
document.write( " <BR>\n" );
document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%5E2%2Bbeta%5E2 ALIGN=MIDDLE  ALT=\"alpha%5E2%2Bbeta%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be easily calculated as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%5E2%2Bbeta%5E2=alpha%5E2%2Bbeta%5E2%2B2%28alpha%29%28beta%29-2%28alpha%29%28beta%29=%28alpha%2Bbeta%29%5E2-2%28alpha%29%28beta%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%5E2%2Bbeta%5E2=%28-3%2F2%29%5E2-2%28-1%2F2%29 ALIGN=MIDDLE  ALT=\"alpha%5E2%2Bbeta%5E2=%28-3%2F2%29%5E2-2%28-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha%5E2%2Bbeta%5E2=9%2F4%2B1 ALIGN=MIDDLE  ALT=\"alpha%5E2%2Bbeta%5E2=9%2F4%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28alpha%5E2%2Bbeta%5E2=13%2F4%29 ALIGN=MIDDLE  ALT=\"highlight%28alpha%5E2%2Bbeta%5E2=13%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28alpha%5E2%2Bbeta%5E2=3%261%2F4%29 ALIGN=MIDDLE  ALT=\"highlight%28alpha%5E2%2Bbeta%5E2=3%261%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28alpha%5E2%2Bbeta%5E2=3.25%29 ALIGN=MIDDLE  ALT=\"highlight%28alpha%5E2%2Bbeta%5E2=3.25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "REASONS:<BR>\n" );
document.write( "The sum of the solutions to the 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;\" > equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2Fa ALIGN=MIDDLE  ALT=\"-b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and the product equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2Fa ALIGN=MIDDLE  ALT=\"c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Why?<BR>\n" );
document.write( "Because if roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+%2B3x-1=0 ALIGN=MIDDLE  ALT=\"2x%5E2+%2B3x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "factoring should transform that equation into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x-alpha%29%28x-beta%29=0 ALIGN=MIDDLE  ALT=\"a%28x-alpha%29%28x-beta%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x%5E2-beta%2Ax-alpha%2Ax%2B%28alpha%29%28beta%29%29=0 ALIGN=MIDDLE  ALT=\"a%28x%5E2-beta%2Ax-alpha%2Ax%2B%28alpha%29%28beta%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x%5E2-%28beta%2Balpha%29%2Ax%2B%28alpha%29%28beta%29%29=0 ALIGN=MIDDLE  ALT=\"a%28x%5E2-%28beta%2Balpha%29%2Ax%2B%28alpha%29%28beta%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2-a%28beta%2Balpha%29%2Ax%2Ba%28alpha%29%28beta%29=0 ALIGN=MIDDLE  ALT=\"ax%5E2-a%28beta%2Balpha%29%2Ax%2Ba%28alpha%29%28beta%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and if that equation must be equivalent to <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;\" >,<BR>\n" );
document.write( "all three coefficients must be the same.<BR>\n" );
document.write( "The coefficient of the term in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-a%28beta%2Balpha%29 ALIGN=MIDDLE  ALT=\"b=-a%28beta%2Balpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2Fa=beta%2Balpha ALIGN=MIDDLE  ALT=\"-b%2Fa=beta%2Balpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The constant (or independent term) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=a%28alpha%29%28beta%29 ALIGN=MIDDLE  ALT=\"c=a%28alpha%29%28beta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2Fa=%28alpha%29%28beta%29 ALIGN=MIDDLE  ALT=\"c%2Fa=%28alpha%29%28beta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );