document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=124200>Question 124200</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2.Suppose that the polynomial f(x) is given f(x) =a.x^2+b.x+c.Suppose ,further,that f(0)=1, f(1)=2, and f(a)=12. Find a, b and c.\r<BR>\n" );
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document.write( " Please help me.help.. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #91127 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=91127\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Since nobody else has responded to this, here's a method you can use, but the solution is<BR>\n" );
document.write( "not a trivial exercise.<BR>\n" );
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document.write( "You are given <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"f%28x%29+=+ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Finding c is easy because f(0) = 1. Substitute 0 for x and the given equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29+=+a%280%5E2%29+%2B+b%280%29+%2B+c ALIGN=MIDDLE  ALT=\"f%280%29+=+a%280%5E2%29+%2B+b%280%29+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and since f(0) = 1 you can say that the right side of this equation equals 1. So:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%280%5E2%29+%2B+b%280%29+%2B+c+=+1 ALIGN=MIDDLE  ALT=\"a%280%5E2%29+%2B+b%280%29+%2B+c+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and this reduces to <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;\" >. That's one answer. You can now replace c with 1 in the given<BR>\n" );
document.write( "equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+ax%5E2+%2B+bx+%2B+1 ALIGN=MIDDLE  ALT=\"f%28x%29+=+ax%5E2+%2B+bx+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next you are told that f(1) = 2. That means that in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+ax%5E2+%2B+bx+%2B+1 ALIGN=MIDDLE  ALT=\"f%28x%29+=+ax%5E2+%2B+bx+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "when you replace x with 1 you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29+=+a%281%5E2%29+%2B+b%281%29+%2B+1+=+2 ALIGN=MIDDLE  ALT=\"f%281%29+=+a%281%5E2%29+%2B+b%281%29+%2B+1+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b+%2B+1+=+2 ALIGN=MIDDLE  ALT=\"a+%2B+b+%2B+1+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve this for b by adding -a - 1 to both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+2+-+a+-+1 ALIGN=MIDDLE  ALT=\"b+=+2+-+a+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Combining the two constants (2 - 1) on the right side reduces the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1+-+a ALIGN=MIDDLE  ALT=\"b+=+1+-+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Remember this result. Next, in the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+ax%5E2+%2B+bx+%2B+1 ALIGN=MIDDLE  ALT=\"f%28x%29+=+ax%5E2+%2B+bx+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "make use of f(a) = 12. If you substitute a for x the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29+=+a%28a%5E2%29+%2B+b%28a%29+%2B+1+=+12 ALIGN=MIDDLE  ALT=\"f%28a%29+=+a%28a%5E2%29+%2B+b%28a%29+%2B+1+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "but recall that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1+-+a ALIGN=MIDDLE  ALT=\"b+=+1+-+a\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Therefore, replace b with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+-+a ALIGN=MIDDLE  ALT=\"1+-+a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29+=+a%28a%5E2%29+%2B+%281-a%29%28a%29+%2B+1+=+12 ALIGN=MIDDLE  ALT=\"f%28a%29+=+a%28a%5E2%29+%2B+%281-a%29%28a%29+%2B+1+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Do the multiplication of the terms in a and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3+%2B+a+-+a%5E2+%2B+1+=+12 ALIGN=MIDDLE  ALT=\"a%5E3+%2B+a+-+a%5E2+%2B+1+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract 12 from both sides and rearrange in descending powers of a. When you do those two<BR>\n" );
document.write( "things you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3+-+a%5E2+%2B+a+-+11+=+0 ALIGN=MIDDLE  ALT=\"a%5E3+-+a%5E2+%2B+a+-+11+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can now solve this equation for a, but the answer you get is not \"nice.\" I question<BR>\n" );
document.write( "whether you copied f(0), f(1), or f(a) correctly since a mistake there would impact the <BR>\n" );
document.write( "answer. However, moving on, I used a calculator to solve this cubic equation for a and got:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+2.43905659 ALIGN=MIDDLE  ALT=\"a+=+2.43905659\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and recall from above that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1+-+a ALIGN=MIDDLE  ALT=\"b+=+1+-+a\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So, substituting 2.43905659 for a results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1+-+2.43905659+=+-1.43905659 ALIGN=MIDDLE  ALT=\"b+=+1+-+2.43905659+=+-1.43905659\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So, we now have as answers: a = 2.43905659 and b = -1.43905659 and c = 1.<BR>\n" );
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document.write( "To check, we substitute these into the original given equation of:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"f%28x%29+=+ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and this equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%282.43905659%29x%5E2+-+%281.43905659%29x+%2B+1 ALIGN=MIDDLE  ALT=\"f%28x%29+=+%282.43905659%29x%5E2+-+%281.43905659%29x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Check that f(0) = 1 by substituting 0 for x to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29+=+%282.43905659%29%280%5E2%29+-+%281.43905659%29%280%29+%2B+1+=+0+-+0+%2B+1+=+1 ALIGN=MIDDLE  ALT=\"f%280%29+=+%282.43905659%29%280%5E2%29+-+%281.43905659%29%280%29+%2B+1+=+0+-+0+%2B+1+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "That checks.<BR>\n" );
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document.write( "Next, check that f(1) = 2 by substuting 1 for x to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+%282.43905659%29%281%5E2%29+-+%281.43905659%29%281%29+%2B+1+=+2.43905659+-+1.43905659+%2B+1+=+1+%2B+1+=+2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "That checks. <BR>\n" );
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document.write( "Next, check that f(a) = 12 by substituting a for x ... that is to say, substituting<BR>\n" );
document.write( "2.43905659 for x to see if the result is 12. Substituting 2.43905659 for x gives you:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282.43905659%29+=+%282.43905659%29%282.43905659%5E2%29+-+%281.43905659%29%282.43905659%29+%2B+1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Doing the multiplication of (2.43905659)(2.43905659^2) and of (1.43905659)(2.43905659) reduces<BR>\n" );
document.write( "the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28a%29+=+f%282.43905659%29+=+14.50994046+-+3.0509940459+%2B+1+=+11+%2B+1+=+12 ALIGN=MIDDLE  ALT=\"+f%28a%29+=+f%282.43905659%29+=+14.50994046+-+3.0509940459+%2B+1+=+11+%2B+1+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and that checks too. <BR>\n" );
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document.write( "So the values found for a, b, and c are correct. The values for a and b are not nice, but<BR>\n" );
document.write( "they work and are therefore correct.<BR>\n" );
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document.write( "Hope this helps ... <BR>\n" );
document.write( ". </SPAN>\n" );
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