document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207458>Question 1207458</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that the real solutions of the equation ax^2 + bx + c = 0 are the reciprocals of the real solutions of the equation cx^2 + bx + a = 0. Assume that b^2-4ac is greater than or equal to 0.\r<BR>\n" );
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document.write( "Do I equate both equations? If not, how is this done?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845337 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=845337\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let r be a root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y+=+ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This will mean <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%5E2%2Bbr%2Bc+=+0 ALIGN=MIDDLE  ALT=\"ar%5E2%2Bbr%2Bc+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the case.<BR>\n" );
document.write( "The input x = r leads to the output y = 0.<BR>\n" );
document.write( "If the root is a real number, then it is where the curve crosses or touches the x axis which we consider it to be an x intercept.\r<BR>\n" );
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document.write( "Divide both sides by r^2<BR>\n" );
document.write( "Do a bit of algebraic rearranging like so<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ar%5E2%2Bbr%2Bc%29%2F%28r%5E2%29+=+0%2F%28r%5E2%29 ALIGN=MIDDLE  ALT=\"%28ar%5E2%2Bbr%2Bc%29%2F%28r%5E2%29+=+0%2F%28r%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ar%5E2%29%2F%28r%5E2%29%2B%28br%29%2F%28r%5E2%29%2Bc%2F%28r%5E2%29+=+0 ALIGN=MIDDLE  ALT=\"%28ar%5E2%29%2F%28r%5E2%29%2B%28br%29%2F%28r%5E2%29%2Bc%2F%28r%5E2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2A%281%2Fr%29%2Bc%2A%281%2F%28r%5E2%29%29+=+0 ALIGN=MIDDLE  ALT=\"a%2Bb%2A%281%2Fr%29%2Bc%2A%281%2F%28r%5E2%29%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2A%281%2Fr%29%5E2%2Bb%2A%281%2Fr%29%2Ba+=+0 ALIGN=MIDDLE  ALT=\"c%2A%281%2Fr%29%5E2%2Bb%2A%281%2Fr%29%2Ba+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This shows that 1/r is a root of y = cx^2+bx+a\r<BR>\n" );
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document.write( "We have proven that r being a root of y = ax^2+bx+c leads to 1/r being a root of y = cx^2+bx+a\r<BR>\n" );
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document.write( "You can reverse this flow of logic to go from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2A%281%2Fr%29%5E2%2Bb%2A%281%2Fr%29%2Ba+=+0 ALIGN=MIDDLE  ALT=\"c%2A%281%2Fr%29%5E2%2Bb%2A%281%2Fr%29%2Ba+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%5E2%2Bbr%2Bc+=+0 ALIGN=MIDDLE  ALT=\"ar%5E2%2Bbr%2Bc+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This will show that 1/r being a root of y = cx^2+bx+a leads to r being a root of y = ax^2+bx+c.\r<BR>\n" );
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document.write( "A rephrasing of this method is found here<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://math.stackexchange.com/questions/2751986/show-that-the-solutions-from-one-quadratic-equation-are-reciprocal-to-the-soluti\">https://math.stackexchange.com/questions/2751986/show-that-the-solutions-from-one-quadratic-equation-are-reciprocal-to-the-soluti</a>\r<BR>\n" );
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document.write( "Another approach\r<BR>\n" );
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document.write( "Let r and s be roots of ax^2+bx+c = 0.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-b+%2B+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"r+=+%28-b+%2B+sqrt%28b%5E2+-+4ac%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=s+=+%28-b+-+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"s+=+%28-b+-+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > due to the quadratic formula.\r<BR>\n" );
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document.write( "t and u are roots of cx^2+bx+a = 0<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%28-b+%2B+sqrt%28b%5E2+-+4ac%29%29%2F%282c%29 ALIGN=MIDDLE  ALT=\"t+=+%28-b+%2B+sqrt%28b%5E2+-+4ac%29%29%2F%282c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+%28-b+-+sqrt%28b%5E2+-+4ac%29%29%2F%282c%29 ALIGN=MIDDLE  ALT=\"u+=+%28-b+-+sqrt%28b%5E2+-+4ac%29%29%2F%282c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > due to the quadratic formula.<BR>\n" );
document.write( "The numerators are the same, but the denominators are now 2c instead of 2a.\r<BR>\n" );
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document.write( "We must require that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C%3E+0 ALIGN=MIDDLE  ALT=\"a+%3C%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+%3C%3E+0 ALIGN=MIDDLE  ALT=\"c+%3C%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to avoid division by zero errors.\r<BR>\n" );
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document.write( "Claim: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fr+=+u ALIGN=MIDDLE  ALT=\"1%2Fr+=+u\"   DISABLED_style= \"max-width:90%; height:auto;\" > i.e. the reciprocal of r is u.\r<BR>\n" );
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document.write( "Proof:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-b+%2B+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"r+=+%28-b+%2B+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fr+=+matrix%281%2C3%2C%22reciprocal%22%2C%22of%22%2C%28-b+%2B+sqrt%28b%5E2+-+4ac%29%29%2F%282a%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fr+=+%282a%29%2F%28-b+%2B+sqrt%28b%5E2+-+4ac%29%29 ALIGN=MIDDLE  ALT=\"1%2Fr+=+%282a%29%2F%28-b+%2B+sqrt%28b%5E2+-+4ac%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fr+=+%282a%29%2F%28-b+%2B+sqrt%28d%29%29 ALIGN=MIDDLE  ALT=\"1%2Fr+=+%282a%29%2F%28-b+%2B+sqrt%28d%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where d = b^2-4ac is the discriminant\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fr+=+%282a%28-b+-+sqrt%28d%29%29%29%2F%28%28-b+%2B+sqrt%28d%29%29%28-b+-+sqrt%28d%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply top and bottom by (-b-sqrt(d)) to rationalize the denominator.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fr+=+%282a%28-b+-+sqrt%28d%29%29%29%2F%28b%5E2-d%29 ALIGN=MIDDLE  ALT=\"1%2Fr+=+%282a%28-b+-+sqrt%28d%29%29%29%2F%28b%5E2-d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Difference of squares rule in the denominator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fr+=+%282a%28-b+-+sqrt%28d%29%29%29%2F%28b%5E2-%28b%5E2-4ac%29%29 ALIGN=MIDDLE  ALT=\"1%2Fr+=+%282a%28-b+-+sqrt%28d%29%29%29%2F%28b%5E2-%28b%5E2-4ac%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fr+=+%282a%28-b+-+sqrt%28d%29%29%29%2F%284ac%29 ALIGN=MIDDLE  ALT=\"1%2Fr+=+%282a%28-b+-+sqrt%28d%29%29%29%2F%284ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fr+=+%28-b+-+sqrt%28d%29%29%2F%282c%29 ALIGN=MIDDLE  ALT=\"1%2Fr+=+%28-b+-+sqrt%28d%29%29%2F%282c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fr+=+%28-b+-+sqrt%28b%5E2-4ac%29%29%2F%282c%29 ALIGN=MIDDLE  ALT=\"1%2Fr+=+%28-b+-+sqrt%28b%5E2-4ac%29%29%2F%282c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fr+=+u ALIGN=MIDDLE  ALT=\"1%2Fr+=+u\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This proves that the reciprocal of r is u.\r<BR>\n" );
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document.write( "Similar steps will show that the reciprocal of s would be t.<BR>\n" );
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document.write( "I'll leave such steps for the student to do.\r<BR>\n" );
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document.write( "Example: <BR>\n" );
document.write( "The roots of x^2+5x+6 = 0 would be r = -2 and s = -3<BR>\n" );
document.write( "The roots of 6x^2+5x+1 = 0 are t = -1/3 and u = -1/2<BR>\n" );
document.write( "This example demonstrates that conditions 1/r = u and 1/s = t are both satisfied.<BR>\n" );
document.write( "I recommend exploring other examples.<BR>\n" );
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