document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1122567>Question 1122567</A>: <I><SPAN STYLE=\"word-break: break-all;\"> True or False? Explain your answer briefly.<BR>\n" );
document.write( "(a) For any real number c , the quadratic equation x^2 + x - c^2 = 0 has two distinct (real) solutions.<BR>\n" );
document.write( "(b) If a > 4 , then the equation ax^2 + 4x + 1 = 0 has no (real) solutions.<BR>\n" );
document.write( "(c) If b^2 &#8722; 4ac &#8805; 0, then the quadratic equation ax^2 + bx + c = 0 has at most one solution.\r<BR>\n" );
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document.write( "Help with this question would be greatly appreciated!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #738695 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=738695\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "For the quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "the discriminant is <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;\" ><br><BR>\n" );
document.write( "It is called the DISCRIMINant because it DISCRIMINates between three cases:<br><BR>\n" );
document.write( "(1) If the discriminant is positive, there are 2 real solutions<BR>\n" );
document.write( "(2) If the discriminant is zero, there is exactly 1 real solution<BR>\n" );
document.write( "(3) If the discriminant is negative, there are no real solutions<br><BR>\n" );
document.write( "(a) The discriminant is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5E2-4%281%29%28-c%5E2%29+=+1%2B4c%5E2 ALIGN=MIDDLE  ALT=\"1%5E2-4%281%29%28-c%5E2%29+=+1%2B4c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  What kinds of values (positive, zero, or negative) can that expression have?<br><BR>\n" );
document.write( "(b) The discriminant is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E2-4%28a%29%281%29+=+16-4a ALIGN=MIDDLE  ALT=\"4%5E2-4%28a%29%281%29+=+16-4a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Given that a > 4, what kinds of values can that expression have?<br><BR>\n" );
document.write( "(c) The answer follows directly from the definition of the discriminant.<br><BR>\n" );
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document.write( "Reply to the student's question....<br><BR>\n" );
document.write( "In the standard form of the quadratic equation, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the \"c\" is whatever the constant term is.<br><BR>\n" );
document.write( "In part (a) of your question, the constant term is \"-c^2\".  That \"c\" is different than the \"c\" in the standard form of the quadratic equation.  It just happens in this case that the constant term contains the variable \"c\".<BR>\n" );
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