document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=967182>Question 967182</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1)Find the smallest integer p such that x^2-2x+p is always greater than 3. Show your working and explain clearly please. Also, can you explain clearly why the graph must be above the x-axis.Help pls.\r<BR>\n" );
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document.write( "2)Find the range of x that satisfies the inequality x^2-6x+9> or = to 0.<BR>\n" );
document.write( "  Please explain and show me the working step by step.\r<BR>\n" );
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document.write( "3)What is the quadratic inequality for the solution which x<-4 and x>8.Expain     clearly and pls show me ur working. Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #591148 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39617)</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=591148\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> #3:\r<BR>\n" );
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document.write( "Quadratic is wanted so that solution set is the union of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C-4 ALIGN=MIDDLE  ALT=\"x%3C-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E8 ALIGN=MIDDLE  ALT=\"x%3E8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Something can be found EITHER for the quadratic being LESS THAN 0,  OR  GREATER THAN 0.  The question is open for you to decide which way you want.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%28x-8%29%3C0 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%28x-8%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has critical values of -4 and 8.  What happens BETWEEN those value?  Pick any x value between -4 and  8 and test it:\r<BR>\n" );
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document.write( "Pick for convenience, x=0.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%2B4%29%280-8%29%3C0 ALIGN=MIDDLE  ALT=\"%280%2B4%29%280-8%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A%28-8%29%3C0 ALIGN=MIDDLE  ALT=\"4%2A%28-8%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-32%3C0 ALIGN=MIDDLE  ALT=\"-32%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This shows that the set of values for x between  -4 and 8 will all make this inequality true; but what we want is not to have THAT as a solution set, because the question was, find the quadratic which has solution set OUTSIDE of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%3Cx%3C8 ALIGN=MIDDLE  ALT=\"-4%3Cx%3C8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "How do we find the requested quadratic expression?  <BR>\n" );
document.write( "How about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%28x-8%29%3E0 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%28x-8%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "Will the solution set be the union of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C-4 ALIGN=MIDDLE  ALT=\"x%3C-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E8 ALIGN=MIDDLE  ALT=\"x%3E8\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
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document.write( "Test those intervals.<BR>\n" );
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document.write( "Pick x=-5, which is within x<-4.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-5%2B4%29%28-5-8%29%3E0 ALIGN=MIDDLE  ALT=\"%28-5%2B4%29%28-5-8%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=%28-1%29%28-13%29%3E0 ALIGN=MIDDLE  ALT=\"%28-1%29%28-13%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=13%3E0 ALIGN=MIDDLE  ALT=\"13%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Pick x=10, which is within x>8.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2A2%3E0 ALIGN=MIDDLE  ALT=\"24%2A2%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48%3E0 ALIGN=MIDDLE  ALT=\"48%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Both parts work.\r<BR>\n" );
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document.write( "There is an answer.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%28x%2B4%29%28x-8%29%3E0%29 ALIGN=MIDDLE  ALT=\"highlight%28%28x%2B4%29%28x-8%29%3E0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "#2:<BR>\n" );
document.write( "Something seems wrong in the question, asking for \"range\" of x.\r<BR>\n" );
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document.write( "#1:<BR>\n" );
document.write( "You might find the use of the discriminant helpful.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x%2Bp%3E3 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2Bp%3E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x%2Bp-3%3E0 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2Bp-3%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "A way to make the quadratic factorable can be  expect p-3=1.<BR>\n" );
document.write( "Then, p=4.<BR>\n" );
document.write( "This would mean  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28x-1%29=x%5E2-2x%2B1%3E0 ALIGN=MIDDLE  ALT=\"%28x-1%29%28x-1%29=x%5E2-2x%2B1%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "What happens at 1?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x%2B%284-1%29%3E0 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2B%284-1%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=x%5E2-2x%2B3%3E0 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2B3%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(x-3)(x+1)>0<BR>\n" );
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document.write( "We should probably want the discriminant to always be negative, so that the quadratic will never cross the x-axis.<BR>\n" );
document.write( "...<BR>\n" );
document.write( "Back to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x%2Bp-3%3E0 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2Bp-3%3E0\"   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=%282%5E2-4%2A%28p-3%29%29%3C0 ALIGN=MIDDLE  ALT=\"%282%5E2-4%2A%28p-3%29%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >------the left member IS the discriminant.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-4p-12%3C0 ALIGN=MIDDLE  ALT=\"4-4p-12%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4p-8%3C0 ALIGN=MIDDLE  ALT=\"-4p-8%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4p%3C8 ALIGN=MIDDLE  ALT=\"-4p%3C8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28p%3E2%29 ALIGN=MIDDLE  ALT=\"highlight%28p%3E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >-----------this makes sure the quadratic does not have x-intercepts.\r<BR>\n" );
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document.write( "An integer is wanted, for p, so that it is AS SMALL as possible.  That would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28p=3%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28p=3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The answer should be checked to be sure. </SPAN>\n" );
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