document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=820654>Question 820654</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello, this is a question from my Algebra 2 textbook: \r<BR>\n" );
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document.write( "Find all values of c for which the equation has (a) two real solutions, (b) one real solution, and (c) two imaginary solutions. \r<BR>\n" );
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document.write( "x^2 - 2x + c = 0\r<BR>\n" );
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document.write( "Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #493658 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=493658\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In the case of x^2 - 2x + c = 0, it is of the form ax^2 + bx + c = 0 where a = 1, b = -2 and c is unknown.\r<BR>\n" );
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document.write( "The equation ax^2 + bx + c = 0 has at most 2 solutions. The type and number of unique solutions are determined by the equation below\r<BR>\n" );
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document.write( "D = b^2 - 4ac\r<BR>\n" );
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document.write( "which is the discriminant formula. The rules are this\r<BR>\n" );
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document.write( "* If D > 0, then you'll have 2 real solutions that are distinct.\r<BR>\n" );
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document.write( "* If D = 0, then you'll have exactly 1 real solution (that's also rational).\r<BR>\n" );
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document.write( "* If D < 0, then you'll have 2 imaginary solutions (aka, complex solutions)\r<BR>\n" );
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document.write( "So from the rules above, we will have 2 real solutions when D > 0.\r<BR>\n" );
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document.write( "D > 0\r<BR>\n" );
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document.write( "b^2 - 4ac > 0  ... replace D with b^2 - 4ac (since D = b^2 - 4ac)\r<BR>\n" );
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document.write( "(-2)^2 - 4(1)c > 0 ... plug in a = 1, b = -2, and leave c alone\r<BR>\n" );
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document.write( "4 - 4c > 0\r<BR>\n" );
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document.write( "4 - 4c + 4c > 0 + 4c\r<BR>\n" );
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document.write( "4 > 4c\r<BR>\n" );
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document.write( "4c < 4\r<BR>\n" );
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document.write( "4c/4 < 4/4\r<BR>\n" );
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document.write( "c < 1\r<BR>\n" );
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document.write( "So if c < 1, then D > 0 which will lead to 2 distinct real solutions.\r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "Now let's move onto the second case: When D = 0\r<BR>\n" );
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document.write( "D = 0\r<BR>\n" );
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document.write( "b^2 - 4ac = 0\r<BR>\n" );
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document.write( "(-2)^2 - 4(1)c = 0\r<BR>\n" );
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document.write( "4 - 4c = 0\r<BR>\n" );
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document.write( "4 = 4c\r<BR>\n" );
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document.write( "4c = 4\r<BR>\n" );
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document.write( "c = 1\r<BR>\n" );
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document.write( "If c = 1, then you'll get exactly one real solution.\r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "You can probably guess what the last answer must be, but let's work through it\r<BR>\n" );
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document.write( "D < 0\r<BR>\n" );
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document.write( "b^2 - 4ac < 0\r<BR>\n" );
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document.write( "(-2)^2 - 4(1)c < 0\r<BR>\n" );
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document.write( "4 - 4c < 0\r<BR>\n" );
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document.write( "4 < 4c\r<BR>\n" );
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document.write( "4c > 4\r<BR>\n" );
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document.write( "c > 1\r<BR>\n" );
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document.write( "So if c > 1, then you'll get 2 imaginary solutions. </SPAN>\n" );
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