document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209641>Question 1209641</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all nonzero constants $a$ such that<BR>\n" );
document.write( "ax^2 + 7x + 2 = 5x^2 + 23x - 12<BR>\n" );
document.write( "has only one distinct root.<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 #849725 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849725\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the values of *a* for which the given quadratic equation has only one distinct root:\r<BR>\n" );
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document.write( "1. **Rewrite the equation in standard form:**\r<BR>\n" );
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document.write( "   Combine like terms to get the equation in the form Ax² + Bx + C = 0:\r<BR>\n" );
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document.write( "   (a - 5)x² + (7 - 23)x + (2 + 12) = 0<BR>\n" );
document.write( "   (a - 5)x² - 16x + 14 = 0\r<BR>\n" );
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document.write( "2. **Apply the discriminant condition:**\r<BR>\n" );
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document.write( "   A quadratic equation has only one distinct root (a double root) when its discriminant (B² - 4AC) is equal to zero.  In our equation:\r<BR>\n" );
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document.write( "   * A = (a - 5)<BR>\n" );
document.write( "   * B = -16<BR>\n" );
document.write( "   * C = 14\r<BR>\n" );
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document.write( "   So, we set the discriminant equal to zero:\r<BR>\n" );
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document.write( "   (-16)² - 4 * (a - 5) * 14 = 0<BR>\n" );
document.write( "   256 - 56(a - 5) = 0<BR>\n" );
document.write( "   256 - 56a + 280 = 0<BR>\n" );
document.write( "   536 - 56a = 0\r<BR>\n" );
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document.write( "3. **Solve for *a*:**\r<BR>\n" );
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document.write( "   56a = 536<BR>\n" );
document.write( "   a = 536 / 56<BR>\n" );
document.write( "   a = 67/7\r<BR>\n" );
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document.write( "Therefore, the only nonzero constant *a* for which the given equation has only one distinct root is a = 67/7.<BR>\n" );
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