document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207406>Question 1207406</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the discriminant to determine whether each quadratic equation has two unequal real solutions,a repeated real solution,or no real solution,without solving the equation.\r<BR>\n" );
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document.write( "x^2 + 4x + 7 = 0\r<BR>\n" );
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document.write( "I know the discriminant is the radicand of the quadratic formula.\r<BR>\n" );
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document.write( "So, I need to use b^2 - 4ac where a = 1, b = 4 and c = 7.\r<BR>\n" );
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document.write( "b^2 - 4ac\r<BR>\n" );
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document.write( "(4)^2 - 4(1)(7)\r<BR>\n" );
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document.write( "16 - 28 = -12\r<BR>\n" );
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document.write( "The answer is -12.\r<BR>\n" );
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document.write( "The discriminant is negative, which means the equation has no real solution.\r<BR>\n" );
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document.write( "You say? \r<BR>\n" );
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document.write( "What would be an example of a repeated real solution or no solution at all? \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845265 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=845265\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Note it is possible, or even probable, that you have not asked the second question correctly.<br><BR>\n" );
document.write( "As implied in the statement of the problem, every quadratic equation has two unequal real solutions, a repeated real solution, or no real solutions.  There is no quadratic equation that has \"no solution at all\".<br><BR>\n" );
document.write( "The given equation has no real solutions: b^2-4ac = 16-28 = -12 < 0.<br><BR>\n" );
document.write( "By definition, any perfect square quadratic polynomial has a repeated real solution (or, better stated, two equal solutions): x^2-6x+9 --> b^2-4ac = 36-36 = 0.<br><BR>\n" );
document.write( "For an example with two real solutions, start with any quadratic polynomial that is a factor of two linear polynomials.  Example: (x+3)(x-2) = x^2+x-6 --> b^2-4ac = 1+24 = 25 > 0.<br><BR>\n" );
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