document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=263310>Question 263310</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The table shows the relationship between x and y in a quadratic equation of the form y = ax^2 + bx + c, where a, b, and c are integers.What is the value of a?\r<BR>\n" );
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document.write( "x     1     2     3     4      5     6      7 \r<BR>\n" );
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document.write( "y    -8     5    24    49     80    117    160 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #194044 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=194044\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> standard form of the equation is equal to :\r<BR>\n" );
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document.write( "y = ax^2 + bx + c\r<BR>\n" );
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document.write( "a,b,c are integers.\r<BR>\n" );
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document.write( "when x = 1, y = -8\r<BR>\n" );
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document.write( "y = ax^2 + bx + c becomes:\r<BR>\n" );
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document.write( "-8 = a*1 + b*1 + c = a + b + c\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "a + b + c = -8\r<BR>\n" );
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document.write( "when x = 2, y = 5\r<BR>\n" );
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document.write( "y = ax^2 + bx + c becomes:\r<BR>\n" );
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document.write( "5 = a*2^2 + b*2 + c = a*4 + b*2 + c\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "4a + 2b + c = 5\r<BR>\n" );
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document.write( "when x = 3, y = 24\r<BR>\n" );
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document.write( "y = ax^2 + bx + c becomes:\r<BR>\n" );
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document.write( "24 = a*3^2 + b*3 + c\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "9a + 3b + c = 24\r<BR>\n" );
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document.write( "you now have 3 equations in 3 unknowns that need to be solved simultaneously.\r<BR>\n" );
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document.write( "those equations are:\r<BR>\n" );
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document.write( "a +   b + c = -8\r\n" );
document.write( "4a + 2b + c =  5\r\n" );
document.write( "9a + 3b + c = 24\r\n" );
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document.write( "solving these equations simultaneously, you get:\r<BR>\n" );
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document.write( "a = 3<BR>\n" );
document.write( "b = 4<BR>\n" );
document.write( "c = -15\r<BR>\n" );
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document.write( "to confirm these answers are correct, substitute these values in the original equations to get:\r<BR>\n" );
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document.write( "a +   b + c = -8\r\n" );
document.write( "4a + 2b + c =  5\r\n" );
document.write( "9a + 3b + c = 24\r\n" );
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document.write( "becomes:\r<BR>\n" );
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document.write( "3    +     4      +    (-15)   =   -8\r\n" );
document.write( "4*3  +     2*4    +    (-15)   =    5\r\n" );
document.write( "9*3  +     3*4    +    (-15)   =   24\r\n" );
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document.write( "simplifying, you get:\r<BR>\n" );
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document.write( " 3   +     4      +    (-15)   =   -8\r\n" );
document.write( "12   +     8      +    (-15)   =    5\r\n" );
document.write( "27   +    12      +    (-15)   =   24\r\n" );
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document.write( "combining like terms, you get:\r<BR>\n" );
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document.write( "     -8   =   -8\r\n" );
document.write( "      5   =    5\r\n" );
document.write( "     24   =   24\r\n" );
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document.write( "this confirms that the values for a,b,c are good.\r<BR>\n" );
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document.write( "plug the values of:\r<BR>\n" );
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document.write( "a = 3<BR>\n" );
document.write( "b = 4<BR>\n" );
document.write( "c = -15\r<BR>\n" );
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document.write( "into the equation of y = ax^2 + bx + c to get:\r<BR>\n" );
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document.write( "y = 3x^2 + 4x - 15\r<BR>\n" );
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document.write( "that's your equation.\r<BR>\n" );
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document.write( "take any (x,y) pair of values you are given and they should be confirmed to be true by plugging them into the equation.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "take (x,y) = (7,160) and plug those values into the equation.\r<BR>\n" );
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document.write( "y = 3x^2 + 4x - 15 becomes:\r<BR>\n" );
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document.write( "160 = 3*(7^2) + 4*7 - 15 which becomes:\r<BR>\n" );
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document.write( "160 = 3*49 + 28 - 15 which becomes:\r<BR>\n" );
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document.write( "160 = 147 + 28 - 15 which becomes:\r<BR>\n" );
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document.write( "160 = 160 which is true confirming the values for a,b,c are good.\r<BR>\n" );
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document.write( "I did not show you how I solved the system of 3 equations in 3 unknowns because it would have detracted from the main focus of what I was trying to show you.\r<BR>\n" );
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document.write( "If you need help with solving this system of 3 equations in 3 unknowns, let me know and I will show you how it was done.\r<BR>\n" );
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document.write( "the question was:\r<BR>\n" );
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document.write( "what is the value of a?\r<BR>\n" );
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document.write( "the answer is:\r<BR>\n" );
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document.write( "the value of a is 3.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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