document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=235135>Question 235135</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help with how to solve this:write a quadratic equation in standard form that passes through these points: (-1, 5), (0,3), (3,9).<BR>\n" );
document.write( "I know it's a systems of equations, but the variables are confusing me... I got<BR>\n" );
document.write( "5 = -A^2 - B + C<BR>\n" );
document.write( "3 = C, and<BR>\n" );
document.write( "9 = 3A^2 + 3B + C\r<BR>\n" );
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document.write( "I tried substituting 3 for c, but I don't know what to do from there<BR>\n" );
document.write( "show work please and explain\r<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 #173295 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=173295\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Recall that each point is of the form (x,y). So for instance, the point (-1, 5) means x=-1 and y=5. This applies for each point given.\r<BR>\n" );
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document.write( "Also, remember that every quadratic can be represented as the equation:\r<BR>\n" );
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document.write( "<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( "where 'a', 'b' and 'c' are real numbers. These values are usually known (and we solve for 'x'), but in this case, we must set up a system of equations to find these values. Note: it turns out that there is only one unique solution to this problem.\r<BR>\n" );
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document.write( "So....\r<BR>\n" );
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document.write( "For the point (-1,5) we know that x=-1 and y=5. Since this point lies on the quadratic, we know that if we plug in x=-1 into the unknown quadratic, we know that we'll get y=5. So the idea is to plug in the given values to find the unknown values.\r<BR>\n" );
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document.write( "Plug these values into the general equation <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;\" > to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=a%28-1%29%5E2%2Bb%28-1%29%2Bc ALIGN=MIDDLE  ALT=\"5=a%28-1%29%5E2%2Bb%28-1%29%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now square -1 to get 1, which means it will absorb into 'a', and simplify:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=a-b%2Bc ALIGN=MIDDLE  ALT=\"5=a-b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the first equation we get is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=a-b%2Bc ALIGN=MIDDLE  ALT=\"5=a-b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Furthermore, since the parabola goes through (0,3) we can plug in x=0 and y=3 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3=a%280%29%5E2%2Bb%280%29%2Bc ALIGN=MIDDLE  ALT=\"3=a%280%29%5E2%2Bb%280%29%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > which simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=3 ALIGN=MIDDLE  ALT=\"c=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Because we've already isolated 'c', we can use this and plug it into the first equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=a-b%2Bc ALIGN=MIDDLE  ALT=\"5=a-b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5=a-b%2B3 ALIGN=MIDDLE  ALT=\"5=a-b%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now solve for 'a' to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=b%2B2 ALIGN=MIDDLE  ALT=\"a=b%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So whatever 'b' is, the value of 'a' will be 2 more than that.\r<BR>\n" );
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document.write( "Finally, we see that the point (3,9) lies on the parabola. So just plug in x=3 and y=9 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=a%283%29%5E2%2Bb%283%29%2Bc ALIGN=MIDDLE  ALT=\"9=a%283%29%5E2%2Bb%283%29%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > and simplify: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=9a%2B3b%2Bc ALIGN=MIDDLE  ALT=\"9=9a%2B3b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From here, we'll use the previously solved for variables 'a' and 'c' to find 'b':\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=9a%2B3b%2Bc ALIGN=MIDDLE  ALT=\"9=9a%2B3b%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=9%28b%2B2%29%2B3b%2B3 ALIGN=MIDDLE  ALT=\"9=9%28b%2B2%29%2B3b%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=b%2B2 ALIGN=MIDDLE  ALT=\"a=b%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=3 ALIGN=MIDDLE  ALT=\"c=3\"   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=9=9b%2B18%2B3b%2B3 ALIGN=MIDDLE  ALT=\"9=9b%2B18%2B3b%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=12b%2B21 ALIGN=MIDDLE  ALT=\"9=12b%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9-21=12b ALIGN=MIDDLE  ALT=\"9-21=12b\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 21 from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12=12b ALIGN=MIDDLE  ALT=\"-12=12b\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12%2F12=b ALIGN=MIDDLE  ALT=\"-12%2F12=b\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 12 to isolate 'b'.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1=b ALIGN=MIDDLE  ALT=\"-1=b\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "So the value of 'b' is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-1 ALIGN=MIDDLE  ALT=\"b=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember that we found that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=b%2B2 ALIGN=MIDDLE  ALT=\"a=b%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=%28-1%29%2B2=1 ALIGN=MIDDLE  ALT=\"a=%28-1%29%2B2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So after everything is said and done, we find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-1 ALIGN=MIDDLE  ALT=\"b=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=3 ALIGN=MIDDLE  ALT=\"c=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving us the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2-x%2B3 ALIGN=MIDDLE  ALT=\"y=x%5E2-x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice how the parabola <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2-x%2B3 ALIGN=MIDDLE  ALT=\"y=x%5E2-x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes through the points (-1, 5), (0,3), and (3,9). So this visually confirms our answer. \r<BR>\n" );
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document.write( "Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2-x%2B3 ALIGN=MIDDLE  ALT=\"y=x%5E2-x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (-1, 5), (0,3), and (3,9) </SPAN>\n" );
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