document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1062373>Question 1062373</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the parabola whose vertex is on the line y=x, axis parallel to x-axis and passing throught (6,-2) and (3,4) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #677309 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39617)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=677309\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The general form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=ay%5E2%2Bby%2Bc ALIGN=MIDDLE  ALT=\"x=ay%5E2%2Bby%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the two given points should allow you to make a linear system of two equations in three unknown variables.  You should be able to eliminate two of them.  Your next equation y=x might allow you to find the value for the last unknown variable.  Converting from general form into standard form might also help.\r<BR>\n" );
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document.write( "The best I have been able to find so far is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%284a%2B2b=-1%2Cx=a%28y-%28b%2F2a%29%29%5E2%2Bc-b%5E2%2F4a%29 ALIGN=MIDDLE  ALT=\"system%284a%2B2b=-1%2Cx=a%28y-%28b%2F2a%29%29%5E2%2Bc-b%5E2%2F4a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <BR>\n" );
document.write( "and the vertex being on line y=x, would be some point  ( <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c-b%5E2%2F%284a%29 ALIGN=MIDDLE  ALT=\"c-b%5E2%2F%284a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2F%282a%29 ALIGN=MIDDLE  ALT=\"b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).\r<BR>\n" );
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document.write( "The linear equation of a and b, and the expected vertex being on line y=x justify a system <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28c-b%5E2%2F%284a%29=b%2F2a%2C4a%2B2b=-1%2C4a-2b%2Bc=6%2C16a%2B4b%2Bc=3%29 ALIGN=MIDDLE  ALT=\"system%28c-b%5E2%2F%284a%29=b%2F2a%2C4a%2B2b=-1%2C4a-2b%2Bc=6%2C16a%2B4b%2Bc=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Still not a final answer.\r<BR>\n" );
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document.write( "The system most likely to give something meaningful might be  the two specific point equations, and the vertex coordinate relationship equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%284a-2b%2Bc=6%2C16a%2B4b%2Bc=3%2C4ac-b%5E2=2b%29 ALIGN=MIDDLE  ALT=\"system%284a-2b%2Bc=6%2C16a%2B4b%2Bc=3%2C4ac-b%5E2=2b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Solve the first of those for c and substitute into the next two equations.  This should be able to make a system<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%284a%2B2b=-1%2C16a%5E2%2Bb%5E2-24a-8ab%2B2b=0%29 ALIGN=MIDDLE  ALT=\"system%284a%2B2b=-1%2C16a%5E2%2Bb%5E2-24a-8ab%2B2b=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Notice that the first equation here has a term  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a ALIGN=MIDDLE  ALT=\"4a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and that a few of the other equation's terms can be factored with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%7D%7D%2C+so+maybe+a+substitution+for+%7B%7B%7B4a ALIGN=MIDDLE  ALT=\"4a%7D%7D%2C+so+maybe+a+substitution+for+%7B%7B%7B4a\"   DISABLED_style= \"max-width:90%; height:auto;\" > might be useful.  This could give a single equation in just the one variable, b.\r<BR>\n" );
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document.write( "Do that and you get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9b%5E2%2B21b%2B7=0 ALIGN=MIDDLE  ALT=\"9b%5E2%2B21b%2B7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Quadratic formula solution will then after simplification give you just for b,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=%28-7%2B-+sqrt%2821%29%29%2F6%29 ALIGN=MIDDLE  ALT=\"highlight%28b=%28-7%2B-+sqrt%2821%29%29%2F6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Still a ways to go to get the possible \"a\" and \"c\" values. </SPAN>\n" );
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