document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196337>Question 1196337</A>: <I><SPAN STYLE=\"word-break: break-all;\"> [ I am reposting because there was a typo in my email when I posted the previous question ]<BR>\n" );
document.write( "Could you please help me with this equation? The solution manual says the answer is x=1 but I am getting x = 1 and x= - 4.<BR>\n" );
document.write( "Could you please help me understand why - 4 is not a solution? I checked it and it doesn't seem to be an extraneous solution. Would be a huge help if you could please include an explanation for the steps.<BR>\n" );
document.write( "Would greatly appreciate your help.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829153 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=829153\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Square both sides to get rid of the square roots. They basically cancel out. <BR>\n" );
document.write( "It's like how multiplication and division are opposites one another.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2%2B5x%29+=+sqrt%282x%2B4%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2%2B5x%29+=+sqrt%282x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28x%5E2%2B5x%29%29%5E2+=+%28sqrt%282x%2B4%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28x%5E2%2B5x%29%29%5E2+=+%28sqrt%282x%2B4%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Squaring both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B5x+=+2x%2B4 ALIGN=MIDDLE  ALT=\"x%5E2%2B5x+=+2x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B5x-2x-4=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B5x-2x-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Get everything to one side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3x-4=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B3x-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%28x-1%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%28x-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factoring\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B4=0 ALIGN=MIDDLE  ALT=\"x%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=0 ALIGN=MIDDLE  ALT=\"x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Zero product property\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-4+ ALIGN=MIDDLE  ALT=\"x+=+-4+\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1 ALIGN=MIDDLE  ALT=\"x+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve each equation for x.\r<BR>\n" );
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document.write( "The two *possible* solutions are <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-4+ ALIGN=MIDDLE  ALT=\"x+=+-4+\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1 ALIGN=MIDDLE  ALT=\"x+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "However, we need to check them both<BR>\n" );
document.write( "Do so in the <font size=4>original equation</font> \r<BR>\n" );
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document.write( "Let's check x = -4<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28-4%29%5E2%2B5%28-4%29%29+=+sqrt%282%28-4%29%2B4%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28-4%29%5E2%2B5%28-4%29%29+=+sqrt%282%28-4%29%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816-20%29+=+sqrt%28-8%2B4%29 ALIGN=MIDDLE  ALT=\"sqrt%2816-20%29+=+sqrt%28-8%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-4%29+=+sqrt%28-4%29 ALIGN=MIDDLE  ALT=\"sqrt%28-4%29+=+sqrt%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This x value leads to the same thing on both sides, but notice how the simplified radicand is negative.\r<BR>\n" );
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document.write( "The domain of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%28x%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=0 ALIGN=MIDDLE  ALT=\"x+%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, i.e. x is nonnegative, when we want y to be a real number. <BR>\n" );
document.write( "So it appears your teacher is ignoring complex/imaginary numbers here. <BR>\n" );
document.write( "You are absolutely correct that x = -4 is a solution but only in the complex number set. <BR>\n" );
document.write( "Regarding real numbers only, we have x = -4 to be extraneous.\r<BR>\n" );
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document.write( "Now check x = 1<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%281%29%5E2%2B5%281%29%29+=+sqrt%282%281%29%2B4%29 ALIGN=MIDDLE  ALT=\"sqrt%28%281%29%5E2%2B5%281%29%29+=+sqrt%282%281%29%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%2B5%29+=+sqrt%282%2B4%29 ALIGN=MIDDLE  ALT=\"sqrt%281%2B5%29+=+sqrt%282%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29+=+sqrt%286%29 ALIGN=MIDDLE  ALT=\"sqrt%286%29+=+sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This time the radicands are positive, so we don't run into the same issue as earlier.<BR>\n" );
document.write( "Both sides are the same real number (approximately 2.449), which confirms x = 1 is the solution.\r<BR>\n" );
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document.write( "If you were to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%28x%5E2%2B5x%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%28x%5E2%2B5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%282x%2B4%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%282x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, as shown in the Desmos link below,<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.desmos.com/calculator/9n9fqdwu9p\">https://www.desmos.com/calculator/9n9fqdwu9p</a><BR>\n" );
document.write( "Notice how the two curves cross at (1, sqrt(6)) and this is the only intersection point. This visually confirms x = 1 as the solution.<BR>\n" );
document.write( "sqrt(6) = 2.449 approximately<BR>\n" );
document.write( "There is no intersection point for when x = -4, which helps rule out this extraneous value.<BR>\n" );
document.write( "In fact, -4 isn't in the domain of either graph.\r<BR>\n" );
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document.write( "Side note: Feel free to use other graphing apps like GeoGebra or a TI83/TI84 if you prefer that better.<BR>\n" );
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