document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181072>Question 1181072</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Which of the following equation has greatest and smallest number of real solutions? <BR>\n" );
document.write( "(a) x^3 + x - 10 = 0<BR>\n" );
document.write( "(b) x^2 + 4x - 15 = 0<BR>\n" );
document.write( "(c) e^x - x = 0<BR>\n" );
document.write( "(d) sec(x) - e^(-x^2)<BR>\n" );
document.write( "Note: Here ^ means power. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #810906 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=810906\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Which of the following equation has greatest and smallest number of real solutions?\r<BR>\n" );
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document.write( "(a) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+%2B+x+-+10+=+0 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+x+-+10+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >...factor\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+%2B2x%5E2-2x%5E2%2B5x-4x+-+10+=+0 ALIGN=MIDDLE  ALT=\"x%5E3+%2B2x%5E2-2x%5E2%2B5x-4x+-+10+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3+-2x%5E2%29%2B%282x%5E2-4x%29+%2B%285x-+10%29+=+0 ALIGN=MIDDLE  ALT=\"%28x%5E3+-2x%5E2%29%2B%282x%5E2-4x%29+%2B%285x-+10%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%28x+-2%29%2B2x%28x-2%29+%2B5%28x-+2%29+=+0 ALIGN=MIDDLE  ALT=\"x%5E2%28x+-2%29%2B2x%28x-2%29+%2B5%28x-+2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+2%29+%28x%5E2+%2B+2x+%2B+5%29+=+0 ALIGN=MIDDLE  ALT=\"%28x+-+2%29+%28x%5E2+%2B+2x+%2B+5%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "one real solution is:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "find two more solutions for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2+%2B+2x+%2B+5%29+=+0 ALIGN=MIDDLE  ALT=\"%28x%5E2+%2B+2x+%2B+5%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "use 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;\" >, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2-4ac%3E0 ALIGN=MIDDLE  ALT=\"+b%5E2-4ac%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > there will be  real solutions\r<BR>\n" );
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document.write( "in your case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=2 ALIGN=MIDDLE  ALT=\"b=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=5 ALIGN=MIDDLE  ALT=\"c=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2%5E2-4%2A1%285%29%3E0 ALIGN=MIDDLE  ALT=\"+2%5E2-4%2A1%285%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4-20%3E0 ALIGN=MIDDLE  ALT=\"+4-20%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-16%3E0 ALIGN=MIDDLE  ALT=\"+-16%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=false ALIGN=MIDDLE  ALT=\"false\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so other two solutions are complex solutions\r<BR>\n" );
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document.write( "(b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+4x+-+15+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+4x+-+15+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "use 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;\" >, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2-4ac%3E0 ALIGN=MIDDLE  ALT=\"+b%5E2-4ac%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > there will be  real solutions\r<BR>\n" );
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document.write( "in your case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=4 ALIGN=MIDDLE  ALT=\"b=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-15 ALIGN=MIDDLE  ALT=\"c=-15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+4%5E2-4%2A1%28-15%29%3E0 ALIGN=MIDDLE  ALT=\"+4%5E2-4%2A1%28-15%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+16%2B60%3E0 ALIGN=MIDDLE  ALT=\"+16%2B60%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+76%3E0 ALIGN=MIDDLE  ALT=\"+76%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >->true, so there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%282%29 ALIGN=MIDDLE  ALT=\"highlight%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > real solutions\r<BR>\n" );
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document.write( "(c) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex+-+x+=+0 ALIGN=MIDDLE  ALT=\"e%5Ex+-+x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex+=x+ ALIGN=MIDDLE  ALT=\"e%5Ex+=x+\"   DISABLED_style= \"max-width:90%; height:auto;\" >....take natural log of both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%5Ex%29+=ln%28x+%29 ALIGN=MIDDLE  ALT=\"ln%28e%5Ex%29+=ln%28x+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Aln%28e%29+=ln%28x+%29 ALIGN=MIDDLE  ALT=\"x%2Aln%28e%29+=ln%28x+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A1+=ln%28x+%29 ALIGN=MIDDLE  ALT=\"x%2A1+=ln%28x+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=ln%28x+%29 ALIGN=MIDDLE  ALT=\"x=ln%28x+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >-><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28No%29 ALIGN=MIDDLE  ALT=\"highlight%28No%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > solution for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(d) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sec%28x%29+-+e%5E%28-x%5E2%29 ALIGN=MIDDLE  ALT=\"sec%28x%29+-+e%5E%28-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "set equal to zero<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sec%28x%29+-+e%5E%28-x%5E2%29=0 ALIGN=MIDDLE  ALT=\"sec%28x%29+-+e%5E%28-x%5E2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sec%28x%29+=e%5E%28-x%5E2%29 ALIGN=MIDDLE  ALT=\"sec%28x%29+=e%5E%28-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> one real solution\r<BR>\n" );
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document.write( "answer: <BR>\n" );
document.write( "the  equation b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+4x+-+15+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+4x+-+15+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has greatest  number of real solutions<BR>\n" );
document.write( "the  equation c) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex+-+x+=+0 ALIGN=MIDDLE  ALT=\"e%5Ex+-+x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has smallest  number of real solutions\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );