document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=884252>Question 884252</A>: <I><SPAN STYLE=\"word-break: break-all;\"> hey! <BR>\n" );
document.write( "I am having a lot of trouble with my year 12 maths and was wondering if anyone was able to assist! <BR>\n" );
document.write( "the problem solving question I have been given with, shows a table of variables being <BR>\n" );
document.write( "x | 0  | 1 | 2 |  3 |  4  |<BR>\n" );
document.write( "y | -3 | 0 | 5 | 12 | 21 |<BR>\n" );
document.write( "for this, it then explains the first and second difference between these numbers <BR>\n" );
document.write( "I understand this as the first difference is<BR>\n" );
document.write( "between -3 and 5 the difference = 3  between and 0 and 5 the difference = 5, between 5 and 12 the difference = 7, and between 12 and 21 the difference = 9\r<BR>\n" );
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document.write( "the second difference then being the difference between these first difference numbers,<BR>\n" );
document.write( "3 and 5 = 2<BR>\n" );
document.write( "5 and 7 = 2<BR>\n" );
document.write( "7 and 9 = 2<BR>\n" );
document.write( "I also understand these are all constant. <BR>\n" );
document.write( "so, the question has given us the rule of y=ax^2 + bx + c which I know is a general function\r<BR>\n" );
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document.write( "it then explains that the first row of entries is obtained by substituting the x values into the rule and said that when x = 1 you just substitute that into the equation. being -> y=a(1) + b(1) + c<BR>\n" );
document.write( "which then simplifies into a + b + c\r<BR>\n" );
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document.write( "the next part then says that the second row of entries (or the first difference in the pattern) is obtained by subtracting consecutive entries in the first row\r<BR>\n" );
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document.write( "my first question being - what and where are the consecutive entries ?<BR>\n" );
document.write( "my second question being that the example then substitutes to <BR>\n" );
document.write( "4a + 2b + c - (a + b + c) = 3a + b \r<BR>\n" );
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document.write( "I am unsure of where these substitutions (being the 4a and 2b) have come from ? \r<BR>\n" );
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document.write( "sorry for the very long explanation but I would be very grateful for an answer! <BR>\n" );
document.write( "thank you for you're time <BR>\n" );
document.write( ":) <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #534118 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=534118\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> we are dealing with a quadratic sequence and we work with the y values<BR>\n" );
document.write( "y | -3 | 0 | 5 | 12 | 21 |<BR>\n" );
document.write( "first difference is  3  5  7  9 (subtract first y value from second y value, second y value from third y value, ....)<BR>\n" );
document.write( "second difference is 2  2  2  2 <BR>\n" );
document.write( "since the constant is 2, we know that we have a n^2 term in the expression for the nth term in the quadratic sequence<BR>\n" );
document.write( "again, we look at the y values<BR>\n" );
document.write( "y | -3 | 0 | 5 | 12 | 21 |<BR>\n" );
document.write( "nth  1   2   3    4    5<BR>\n" );
document.write( "n^2  1   4   9   16   25<BR>\n" );
document.write( "subtract n^2 from y values<BR>\n" );
document.write( "    -4  -4  -4   -4   -4<BR>\n" );
document.write( "the nth term in the geometric sequence is n^2 - 4<BR>\n" );
document.write( "we can work this from another direction, we know<BR>\n" );
document.write( "-3 = a(1^2) +b(1) +c<BR>\n" );
document.write( "0 =  a(0^2) +b(0) +c <BR>\n" );
document.write( "5 =  a(5^2) +b(5) +c<BR>\n" );
document.write( "or<BR>\n" );
document.write( "-3 = a +b +c<BR>\n" );
document.write( "0 = c<BR>\n" );
document.write( "5 = 25a + 5b +c<BR>\n" );
document.write( "we know c = 0<BR>\n" );
document.write( "-3 = a +b<BR>\n" );
document.write( "1 = 5a +b<BR>\n" );
document.write( "this can be solved for a and b<BR>\n" );
document.write( "a = 1, b = -4<BR>\n" );
document.write( "y = x^2 -4x is the quadratic form<BR>\n" );
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