document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=567835>Question 567835</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you get the quadratic equation if you are given a table of values? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #366726 by </B><A HREF=/tutors/aboutme.mpl?userid=issacodegard><B>issacodegard(60)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=issacodegard><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=issacodegard><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=366726\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You must be given at least 3 points, call them (x1,y1),(x2,y2),(x3,y3).  Then we know the equation has the form f(x)=ax^2+bx+c.  We get a system of 3 equations in 3 unknowns.  Solving for a,b,c in,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x1%29=a%28x1%29%5E2%2Bb%28x1%29%2Bc=y1 ALIGN=MIDDLE  ALT=\"f%28x1%29=a%28x1%29%5E2%2Bb%28x1%29%2Bc=y1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x2%29=a%28x2%29%5E2%2Bb%28x2%29%2Bc=y2 ALIGN=MIDDLE  ALT=\"f%28x2%29=a%28x2%29%5E2%2Bb%28x2%29%2Bc=y2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x3%29=a%28x3%29%5E2%2Bb%28x3%29%2Bc=y3 ALIGN=MIDDLE  ALT=\"f%28x3%29=a%28x3%29%5E2%2Bb%28x3%29%2Bc=y3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "gives you the equation for f(x).\r<BR>\n" );
document.write( "\n" );
document.write( "I'll give an example,<BR>\n" );
document.write( "If we know that three points are (-1,-2),(1,4),(2,13).  Then we set up three equations about a,b,c that we know:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-1%29=a%28-1%29%5E2%2Bb%28-1%29%2Bc=-2 ALIGN=MIDDLE  ALT=\"f%28-1%29=a%28-1%29%5E2%2Bb%28-1%29%2Bc=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=a%281%29%5E2%2Bb%281%29%2Bc=4 ALIGN=MIDDLE  ALT=\"f%281%29=a%281%29%5E2%2Bb%281%29%2Bc=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%29=a%282%29%5E2%2Bb%282%29%2Bc=13 ALIGN=MIDDLE  ALT=\"f%282%29=a%282%29%5E2%2Bb%282%29%2Bc=13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So we have,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-b%2Bc=-2 ALIGN=MIDDLE  ALT=\"a-b%2Bc=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc=4 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2B2b%2Bc=13 ALIGN=MIDDLE  ALT=\"4a%2B2b%2Bc=13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We need to solve for a,b,c.  So, we subtract eqn1 from eqn2 to get,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-b%2Bc=-2 ALIGN=MIDDLE  ALT=\"a-b%2Bc=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b=6 ALIGN=MIDDLE  ALT=\"2b=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2B2b%2Bc=13 ALIGN=MIDDLE  ALT=\"4a%2B2b%2Bc=13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide eqn2 by 2 to get b=3.  Then, we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-3%2Bc=-2 ALIGN=MIDDLE  ALT=\"a-3%2Bc=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2B2%283%29%2Bc=13 ALIGN=MIDDLE  ALT=\"4a%2B2%283%29%2Bc=13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bc=1 ALIGN=MIDDLE  ALT=\"a%2Bc=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2Bc=7 ALIGN=MIDDLE  ALT=\"4a%2Bc=7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtract the first eqn from the second to get,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bc=1 ALIGN=MIDDLE  ALT=\"a%2Bc=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a=6 ALIGN=MIDDLE  ALT=\"3a=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide the second eqn by 3 to get a=2.  Then we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Bc=1 ALIGN=MIDDLE  ALT=\"2%2Bc=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-1 ALIGN=MIDDLE  ALT=\"c=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore we find that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E2%2B3x-1 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E2%2B3x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );