document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209288>Question 1209288</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the number of integers n that satisfy n^2 < 144 + 24n. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848109 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=848109\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "For now I'll consider the equation n^2 = 144 + 24n\r<BR>\n" );
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document.write( "Let's get everything to one side and then apply the quadratic formula.<BR>\n" );
document.write( "n^2 = 144 + 24n<BR>\n" );
document.write( "n^2-24n-144 = 0<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"n+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   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=n+=+%28-%28-24%29%2B-sqrt%28%28-24%29%5E2-4%281%29%28-144%29%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"n+=+%28-%28-24%29%2B-sqrt%28%28-24%29%5E2-4%281%29%28-144%29%29%29%2F%282%281%29%29\"   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=n+=+%2824%2B-sqrt%281152%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%2824%2B-sqrt%281152%29%29%2F%282%29\"   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=n+=+%2824%2Bsqrt%281152%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%2824%2Bsqrt%281152%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%2824-sqrt%281152%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"n+=+%2824-sqrt%281152%29%29%2F%282%29\"   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=n+=+28.970563 ALIGN=MIDDLE  ALT=\"n+=+28.970563\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+-4.970563 ALIGN=MIDDLE  ALT=\"n+=+-4.970563\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The decimal values are approximate.\r<BR>\n" );
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document.write( "Draw out a number line. <BR>\n" );
document.write( "Mark -4.970563 and 28.970563 on it.\r<BR>\n" );
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document.write( "Let's test an integer value to the left of -4.970563<BR>\n" );
document.write( "I'll try n = -5<BR>\n" );
document.write( "n^2 < 144 + 24n<BR>\n" );
document.write( "(-5)^2 < 144 + 24*(-5)<BR>\n" );
document.write( "25 < 24<BR>\n" );
document.write( "which is false<BR>\n" );
document.write( "We eliminate the subset of integers that are -5 or smaller.\r<BR>\n" );
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document.write( "Now try an integer value between -4.970563 and 28.970563<BR>\n" );
document.write( "I'll pick n = 0<BR>\n" );
document.write( "(0)^2 < 144 + 24(0)<BR>\n" );
document.write( "0 < 144<BR>\n" );
document.write( "which is true<BR>\n" );
document.write( "If n is any of the following integers {-4, -3, ..., 27, 28} then the inequality n^2 < 144 + 24n is true.\r<BR>\n" );
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document.write( "Lastly we need to test an integer larger than 28.970563<BR>\n" );
document.write( "I'll pick n = 29<BR>\n" );
document.write( "n^2 < 144 + 24n<BR>\n" );
document.write( "(29)^2 < 144 + 24*(29)<BR>\n" );
document.write( "841 < 840<BR>\n" );
document.write( "which is false<BR>\n" );
document.write( "We can eliminate the set of integers that are 29 or larger\r<BR>\n" );
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document.write( "To summarize we found the integer solution set is {-4, -3, ..., 27, 28}<BR>\n" );
document.write( "There are p = 4 items in {-4, -3, -2, -1}<BR>\n" );
document.write( "There are q = 28 items in {1,2,3,...,27,28}<BR>\n" );
document.write( "So far that's p+q = 4+28 = 32 nonzero integers accounted for.<BR>\n" );
document.write( "Then there's 0 itself to bump the final count to 33.\r<BR>\n" );
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document.write( "Or you can use the formula last-first+1 to count the number of consecutive integers in a list.<BR>\n" );
document.write( "last-first+1 = 28-(-4)+1 = 28+4+1 = 33\r<BR>\n" );
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document.write( "Answer: 33<BR>\n" );
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