document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=16536>Question 16536</A>: <I><SPAN STYLE=\"word-break: break-all;\">  if (a+2)x^2- 2ax - a = 0<BR>\n" );
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document.write( "what is range of a ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #8082 by </B><A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><B>AnlytcPhil(1810)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><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=AnlytcPhil><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=8082\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face = \"courier new\"><BR>\n" );
document.write( "if (a+2)x^2- 2ax - a = 0<BR>\n" );
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document.write( "what is range of a ?<BR>\n" );
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document.write( "If you were to solve this problem by the quadratic formula, <BR>\n" );
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document.write( "A = (a+2), B = -2a, C = -a<BR>\n" );
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document.write( "the expression underneath the square root, B2 - 4AC, known as the<BR>\n" );
document.write( "'discriminant', must not be negative.<BR>\n" );
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document.write( "` ` ` ` ` `B2 - 4AC >= 0<BR>\n" );
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document.write( "(-2a)2 - 4(a+2)(-a) >= 0<BR>\n" );
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document.write( "` ` ` 4a2 + 4a(a+2) >= 0<BR>\n" );
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document.write( "` ` `4a2 + 4a2 + 8a >= 0<BR>\n" );
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document.write( "` ` ` ` ` `8a2 + 8a >= 0<BR>\n" );
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document.write( "` ` ` ` ` 8a(a + 1) >= 0<BR>\n" );
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document.write( "The zeros of the left side are 0 and -1<BR>\n" );
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document.write( "Make a number line, and mark these solid since they are solutions. <BR>\n" );
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document.write( "-4 -3 -2 -1 `0 +1 +2 +3 +4<BR>\n" );
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document.write( "Choose any value in the leftmost region, the region left of -1.  The easiest<BR>\n" );
document.write( "value to choose is -2.  Substitute it into the expression<BR>\n" );
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document.write( "8a(a + 1) <BR>\n" );
document.write( "8(-2)(-2 + 1) = -16(-1) = +16. This is nonnegative, so shade all those values.<BR>\n" );
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document.write( "`-4 -3 -2 -1  0 +1 +2 +3 +4<BR>\n" );
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document.write( "Choose any value in the middle region, the region between -1 and 0.  The<BR>\n" );
document.write( "easiest value to choose is -1/2.  Substitute it into the expression<BR>\n" );
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document.write( "8a(a + 1) <BR>\n" );
document.write( "8(-1/2)(-1/2 + 1) = -4(1/2) = -2. This is negative, so do not shade the values<BR>\n" );
document.write( "in the middle region. <BR>\n" );
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document.write( "Choose any value in the rightmost region, the region right of 0.  The easiest<BR>\n" );
document.write( "value to choose is 1.  Substitute it into the expression<BR>\n" );
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document.write( "8a(a + 1) <BR>\n" );
document.write( "8(1)(1 + 1) = 8(2) = +16. This is nonnegative, so shade all those values.<BR>\n" );
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document.write( "`-4 -3 -2 -1 `0 +1 +2 +3 +4<BR>\n" );
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document.write( "The interval notation for this is<BR>\n" );
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document.write( "(-<font face = \"symbol\">¥</font>,-1] <font face = \"arial\">U</font> [0, <font face = \"symbol\">¥</font>)<BR>\n" );
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document.write( "Edwin<BR>\n" );
document.write( "AnlytcPhil@aol.com </SPAN>\n" );
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