document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=948575>Question 948575</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me answer this: show that m(x^2-x)+1-m=x^2 has real roots for all real values of m. (P.S. Do forgive me for all the inconvenience.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #579002 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39626)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=579002\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%28x%5E2-x%29%2B1-m=x%5E2 ALIGN=MIDDLE  ALT=\"m%28x%5E2-x%29%2B1-m=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Quadratic equation with \"unknown\" x, and constant m.<BR>\n" );
document.write( "Real roots?  Discriminant must be non-negative.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mx%5E2-mx%2B1-m-x%5E2=0 ALIGN=MIDDLE  ALT=\"mx%5E2-mx%2B1-m-x%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mx%5E2-x%5E2-mx%2B1-m=0 ALIGN=MIDDLE  ALT=\"mx%5E2-x%5E2-mx%2B1-m=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m-1%29x%5E2-mx%2B%281-m%29=0 ALIGN=MIDDLE  ALT=\"%28m-1%29x%5E2-mx%2B%281-m%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the given equation now in general form.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Inequality for the discriminant needed:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28m%5E2-4%28m-1%29%281-m%29%3E=0%29 ALIGN=MIDDLE  ALT=\"highlight_green%28m%5E2-4%28m-1%29%281-m%29%3E=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, left member is the discriminant.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-4%28m-1%29%28-1%29%28m-1%29%3E=0 ALIGN=MIDDLE  ALT=\"m%5E2-4%28m-1%29%28-1%29%28m-1%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2%2B4%28m-1%29%5E2%3E=0 ALIGN=MIDDLE  ALT=\"m%5E2%2B4%28m-1%29%5E2%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2%2B4%28m%5E2-2m%2B1%29%3E=0 ALIGN=MIDDLE  ALT=\"m%5E2%2B4%28m%5E2-2m%2B1%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2%2B4m%5E2-8m%2B4%3E=0 ALIGN=MIDDLE  ALT=\"m%5E2%2B4m%5E2-8m%2B4%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%285m%5E2-8m%2B4%3E=0%29 ALIGN=MIDDLE  ALT=\"highlight_green%285m%5E2-8m%2B4%3E=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The truth of this m inequality depends on no real roots for m, meaning the discriminant of just the m inequality must be NEGATIVE.<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-8%29%5E2-4%2A5%284%29%3C0 ALIGN=MIDDLE  ALT=\"%28-8%29%5E2-4%2A5%284%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the m quadratic inequality has no real roots.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64-20%2A4%3C0 ALIGN=MIDDLE  ALT=\"64-20%2A4%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64-80%3C0 ALIGN=MIDDLE  ALT=\"64-80%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28-16%3C0%29 ALIGN=MIDDLE  ALT=\"highlight_green%28-16%3C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >------TRUE.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "That truth means that the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5m%5E2-8m%2B4%3E=0 ALIGN=MIDDLE  ALT=\"5m%5E2-8m%2B4%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  is satisfied; and that means that the original given equation will have real roots (for all m). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );