document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=74399>Question 74399</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0130443980/algebrahomeworkh>Prentice Hall Algebra 1 Practice Workbook</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> 1. Twenty more than the square of a number is twelve times the number. find the numbers.<BR>\n" );
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document.write( "2. The Product of three more than a number times two less than the number is fourteen. Find the numbers.<BR>\n" );
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document.write( "3. The sum of the squares of two consecutive odd positive integers is 202. Find the numbers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #53410 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=53410\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Problem 1: Twenty more than the square of a number is twelve times the number. find the numbers.<BR>\n" );
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document.write( "Begin by calling the unknown number or numbers N.  Squaring N gives you <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2 ALIGN=MIDDLE  ALT=\"N%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <BR>\n" );
document.write( "20 more than that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2+%2B+20 ALIGN=MIDDLE  ALT=\"N%5E2+%2B+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The problem then tells you that this equals 12 times <BR>\n" );
document.write( "the number which is written as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12N ALIGN=MIDDLE  ALT=\"12N\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  So the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2+%2B+20+=+12N ALIGN=MIDDLE  ALT=\"N%5E2+%2B+20+=+12N\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get this into the standard form of a quadratic equation by subtracting 12N from both sides.<BR>\n" );
document.write( "The result is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2+-+12N+%2B+20+=+0 ALIGN=MIDDLE  ALT=\"N%5E2+-+12N+%2B+20+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The left side of this equation can be factored into two terms as follows:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28N-10%29%2A%28N-2%29+=+0 ALIGN=MIDDLE  ALT=\"%28N-10%29%2A%28N-2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "(You can check this by multiplying these two terms and seeing that they get you back to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2+-+12N+%2B+20 ALIGN=MIDDLE  ALT=\"N%5E2+-+12N+%2B+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >.)<BR>\n" );
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document.write( "Notice that the factored form of the equation will be true if either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28N-10%29+=+0 ALIGN=MIDDLE  ALT=\"%28N-10%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or if<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28N-2%29+=+0 ALIGN=MIDDLE  ALT=\"%28N-2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > because if either factor equals zero, the entire left side of the equation<BR>\n" );
document.write( "equals zero.<BR>\n" );
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document.write( "So, one at a time set the two factors equal to zero and solve for N.  From the first factor<BR>\n" );
document.write( "you will find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N=10 ALIGN=MIDDLE  ALT=\"N=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and from the second factor you will find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N=2 ALIGN=MIDDLE  ALT=\"N=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So your problem has two answers that work ... N = 10 and N = 2.  Check these two out by<BR>\n" );
document.write( "returning to the problem and asking yourself, \"Does 20 more than the square of 10 equal<BR>\n" );
document.write( "12 times 10?\" This simplifies to, \"Does 20 more than 100 equal 120?\" and, of course, it<BR>\n" );
document.write( "does.  Then you can do the same with 2.  \"Does 20 more than the square of 2 equal<BR>\n" );
document.write( "12 times 2?\" This simplifies to, \"Does 20 more than 4 equal 24?\" and, of course, it does also.  <BR>\n" );
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document.write( "Problem 2: The Product of three more than a number times two less than the number is <BR>\n" );
document.write( "fourteen. Find the numbers.<BR>\n" );
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document.write( "Again, let N represent the unknown number or numbers.  Three more than that number is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28N%2B3%29 ALIGN=MIDDLE  ALT=\"%28N%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and two less than that number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28N-2%29 ALIGN=MIDDLE  ALT=\"%28N-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The problem says that product of <BR>\n" );
document.write( "these two quantities equals 14. In equation form this becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28N%2B3%29%2A%28N-2%29+=+14 ALIGN=MIDDLE  ALT=\"%28N%2B3%29%2A%28N-2%29+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply out the left side and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2+%2B+N+-+6+=+14 ALIGN=MIDDLE  ALT=\"N%5E2+%2B+N+-+6+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Put this into standard quadratic form by subtracting 14 from both sides to eliminate<BR>\n" );
document.write( "the 14 on the right side of the equation.  The result of this subtraction from both sides is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2+%2B+N+-20+=+0 ALIGN=MIDDLE  ALT=\"N%5E2+%2B+N+-20+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The left side of this equation factors and the factored form is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28N%2B5%29%2A%28N-4%29+=+0 ALIGN=MIDDLE  ALT=\"%28N%2B5%29%2A%28N-4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Just as we did in the first problem (and for the same reason) set each of the factors equal<BR>\n" );
document.write( "to zero and you find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+=+-5 ALIGN=MIDDLE  ALT=\"N+=+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+=+%2B4 ALIGN=MIDDLE  ALT=\"N+=+%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the two solutions.    <BR>\n" );
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document.write( "Check by determining if (-5+3)*(-5-2) equals 14.  (It does). And then determine if <BR>\n" );
document.write( "(4+3)*(4-2) equals 14.  (It does also).<BR>\n" );
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document.write( "Problem 3: The sum of the squares of two consecutive odd positive integers is 202. Find <BR>\n" );
document.write( "the numbers.<BR>\n" );
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document.write( "Let N represent the odd positive integer.  Then the next odd positive integer must be <BR>\n" );
document.write( "N+2 because odd integers are always separated by an even integer in between. [Think that<BR>\n" );
document.write( "two odd integers in a row are 3 and 5 or 7 and 9.  They are always 2 apart.]<BR>\n" );
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document.write( "The squares of these two integers are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2 ALIGN=MIDDLE  ALT=\"N%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28N%2B2%29%5E2 ALIGN=MIDDLE  ALT=\"%28N%2B2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is equal to <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2+%2B+4N+%2B+4 ALIGN=MIDDLE  ALT=\"N%5E2+%2B+4N+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. When you add these two squares together you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2+%2B+N%5E2+%2B+4N+%2B+4 ALIGN=MIDDLE  ALT=\"N%5E2+%2B+N%5E2+%2B+4N+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and by combining the squared terms this becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2N%5E2+%2B+4N+%2B+4 ALIGN=MIDDLE  ALT=\"2N%5E2+%2B+4N+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Set this sum of the squares equal to 202 as required by the problem.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2N%5E2+%2B+4N+%2B+4+=+202 ALIGN=MIDDLE  ALT=\"2N%5E2+%2B+4N+%2B+4+=+202\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then get this into standard quadratic form by subtracting 202 from both sides to eliminate the<BR>\n" );
document.write( "202 on the right side and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2N%5E2+%2B+4N+-+198+=+0 ALIGN=MIDDLE  ALT=\"2N%5E2+%2B+4N+-+198+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Simplify the problem a little by dividing both sides (all terms) by 2 to eliminate the <BR>\n" );
document.write( "multiplier of 2 on the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2 ALIGN=MIDDLE  ALT=\"N%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term and the equation reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%5E2+%2B+2N+-+99+=+0 ALIGN=MIDDLE  ALT=\"N%5E2+%2B+2N+-+99+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This factors to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28N+%2B+11%29%2A%28N+-+9%29+=+0 ALIGN=MIDDLE  ALT=\"%28N+%2B+11%29%2A%28N+-+9%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Setting each factor equal to zero as was done in the other problems above results in<BR>\n" );
document.write( "two answers:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+=+-11 ALIGN=MIDDLE  ALT=\"N+=+-11\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N+=+9 ALIGN=MIDDLE  ALT=\"N+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The problem, however, requires the answer to<BR>\n" );
document.write( "be a positive odd number, so we ignore the negative answer for N and the answer to this <BR>\n" );
document.write( "problem is that N = +9.  The other positive odd integer is 2 plus the +9 or 11.  In summary<BR>\n" );
document.write( "the two odd positive integers you are looking for are 9 and 11.<BR>\n" );
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document.write( "Check by finding the sum of their squares.  The squares are 81 and 121 and their sum is 202<BR>\n" );
document.write( "as is required by the problem, so the answers we got check.<BR>\n" );
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document.write( "Hope that in these three problems you see how to work with setting up these quadratic<BR>\n" );
document.write( "equations and you see how you can solve them by factoring, if they do factor. If they do not<BR>\n" );
document.write( "factor you will have to use another method to solve them, and the most universal method is<BR>\n" );
document.write( "the quadratic formula which is another lesson entirely.<BR>\n" );
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