document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207450>Question 1207450</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that the product of the roots of a quadratic equation is c/a.\r<BR>\n" );
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document.write( "I see the word product in the application. This tells me to multiply two quadratic formulas and simplify to get c/a. \r<BR>\n" );
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document.write( "Yes?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845319 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=845319\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "a,b,c are real numbers<BR>\n" );
document.write( "'a' is nonzero\r<BR>\n" );
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document.write( "The standard template for a quadratic is<BR>\n" );
document.write( "ax^2 + bx + c = 0<BR>\n" );
document.write( "Divide everything by 'a' to get<BR>\n" );
document.write( "x^2 + (b/a)x + c/a = 0<BR>\n" );
document.write( "We'll come back to this later.<BR>\n" );
document.write( "The key thing to pay attention to is that the leading coefficient is 1.\r<BR>\n" );
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document.write( "Let r and s be the two roots of this quadratic.<BR>\n" );
document.write( "If x = r is a root then x-r is a factor<BR>\n" );
document.write( "Same goes for the other root.<BR>\n" );
document.write( "We find that:<BR>\n" );
document.write( "(x-r)(x-s) = 0<BR>\n" );
document.write( "which rewrites to<BR>\n" );
document.write( "x^2-(r+s)x+rs = 0<BR>\n" );
document.write( "Here the leading coefficient is 1 as well.\r<BR>\n" );
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document.write( "The product of the roots is the \"rs\" term at the end.<BR>\n" );
document.write( "Compare that to the last term in x^2 + (b/a)x + c/a = 0 to see it matches with the c/a.<BR>\n" );
document.write( "Both are constant terms not attached to x in any fashion.\r<BR>\n" );
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document.write( "Therefore, rs = c/a\r<BR>\n" );
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document.write( "For more information, search out \"Vieta's Formulas\".\r<BR>\n" );
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document.write( "Alternative route\r<BR>\n" );
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document.write( "You can use the quadratic formula like tutor Theo has done, but it could get a bit messy.<BR>\n" );
document.write( "The good news is that notice how d = b^2 - 4ac is the discriminant so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > would become <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2B-sqrt%28d%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2B-sqrt%28d%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then that would lead to these roots<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2Bsqrt%28d%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2Bsqrt%28d%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b-sqrt%28d%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b-sqrt%28d%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The numerators multiply to b^2-d when using the difference of squares rule. <BR>\n" );
document.write( "Then compute b^2-d = b^2-(b^2-4ac) = 4ac<BR>\n" );
document.write( "The denominators multiply to 4a^2\r<BR>\n" );
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document.write( "So this is a fairly quick way to arrive at (4ac)/(4a^2) = c/a.<BR>\n" );
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