document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=90898>Question 90898</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2-5y=0 ALIGN=MIDDLE  ALT=\"y%5E2-5y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >   Solve the equation first by completing the square and then by factoring. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #65957 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=65957\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2-5y=0 ALIGN=MIDDLE  ALT=\"y%5E2-5y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "Solve this by completing the square and then by factoring.<BR>\n" );
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document.write( "Let's use the factoring method first. It is so easy, and we can use the answers we get to<BR>\n" );
document.write( "check the results we get from completing the square.<BR>\n" );
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document.write( "In the factoring method, you can see that y is a factor of both terms on the left side.<BR>\n" );
document.write( "Factor a y 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=y%2A%28y+-+5%29+=+0 ALIGN=MIDDLE  ALT=\"y%2A%28y+-+5%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now notice that this equation will be true if either of the factors equals zero because<BR>\n" );
document.write( "a multiplication by zero on the left side will make the left side be zero and therefore<BR>\n" );
document.write( "it will equal the right side.<BR>\n" );
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document.write( "So we set each of the factors equal to zero ...<BR>\n" );
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document.write( "First y = 0. No solving is necessary. That is one answer for y.<BR>\n" );
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document.write( "Next y - 5 = 0. If you add 5 to both sides this becomes y = 5. And this is the second<BR>\n" );
document.write( "answer for y.  <BR>\n" );
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document.write( "Now let's try completing the square.<BR>\n" );
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document.write( "Start with<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+-5y+=+0 ALIGN=MIDDLE  ALT=\"y%5E2+-5y+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Take half of the multiplier of y. The multiplier is -5 and half of that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2F2 ALIGN=MIDDLE  ALT=\"-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Square this to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2B25%2F4 ALIGN=MIDDLE  ALT=\"%2B25%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is the number that you need to add to both sides<BR>\n" );
document.write( "of the equation to make the left side a perfect square.  When you add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%2F4 ALIGN=MIDDLE  ALT=\"25%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both <BR>\n" );
document.write( "sides of the equation you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+-5y+%2B+25%2F4+=+25%2F4 ALIGN=MIDDLE  ALT=\"y%5E2+-5y+%2B+25%2F4+=+25%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The left side of this equation is a perfect square. The quantity that squares to this left<BR>\n" );
document.write( "side is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y+-5%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28y+-5%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and the equation then becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y+-+5%2F2%29%5E2+=+25%2F4 ALIGN=MIDDLE  ALT=\"%28y+-+5%2F2%29%5E2+=+25%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now take the square root of both sides and recognize that the right side will have two<BR>\n" );
document.write( "possibilities ... one with a plus sign and one with a minus sign. Therefore, after taking<BR>\n" );
document.write( "the square root of both sides you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+-+5%2F2+=+%2B5%2F2 ALIGN=MIDDLE  ALT=\"y+-+5%2F2+=+%2B5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+-+5%2F2+=+-5%2F2 ALIGN=MIDDLE  ALT=\"y+-+5%2F2+=+-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can solve for y in both of these equations by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F2 ALIGN=MIDDLE  ALT=\"5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides. In<BR>\n" );
document.write( "the first of these when you add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2B5%2F2 ALIGN=MIDDLE  ALT=\"%2B5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides then:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+-+5%2F2+=+5%2F2 ALIGN=MIDDLE  ALT=\"y+-+5%2F2+=+5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "becomes<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+5%2F2+%2B+5%2F2+=+10%2F2+=+5 ALIGN=MIDDLE  ALT=\"y+=+5%2F2+%2B+5%2F2+=+10%2F2+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and the other equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+-+5%2F2+=+-5%2F2 ALIGN=MIDDLE  ALT=\"y+-+5%2F2+=+-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ... when you add +<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F2 ALIGN=MIDDLE  ALT=\"5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides<BR>\n" );
document.write( "becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+0 ALIGN=MIDDLE  ALT=\"y+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > because adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2B5%2F2 ALIGN=MIDDLE  ALT=\"%2B5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides cancels the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2F2 ALIGN=MIDDLE  ALT=\"-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > on<BR>\n" );
document.write( "both sides.<BR>\n" );
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document.write( "These two answers agree with the answers that we got by factoring, but factoring is a <BR>\n" );
document.write( "much quicker way to do the problem with less chance for making an error.<BR>\n" );
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document.write( "Hope that this helps you to understand the problem and to recognize that if you can factor<BR>\n" );
document.write( "the left side, it is generally easier than completing the square or using the quadratic<BR>\n" );
document.write( "formula for that matter. </SPAN>\n" );
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