document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=720317>Question 720317</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Here is my problem.\r<BR>\n" );
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document.write( "[SQRT(x + 7)] - 2[SQRT(x)] =-2\r<BR>\n" );
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document.write( "OR\r<BR>\n" );
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document.write( "&#8730;(x + 7) - 2&#8730;(x) = -2 \r<BR>\n" );
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document.write( "Now, the steps I took to solve this problem are to first square both sides:\r<BR>\n" );
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document.write( "[&#8730;(x + 7) - 2&#8730;(x)] * [&#8730;(x + 7) - 2&#8730;(x)] = 4\r<BR>\n" );
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document.write( "so then I FOIL the left side, resulting in:\r<BR>\n" );
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document.write( "(x+7) - [&#8730;(x + 7) * - 2&#8730;(x)] - [ - 2&#8730;(x) * &#8730;(x + 7)] + 4x\r<BR>\n" );
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document.write( "So then I thought to subtract (x+7) and (4x) to both sides\r<BR>\n" );
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document.write( " - [&#8730;(x + 7) * - 2&#8730;(x)] - [ - 2&#8730;(x) * &#8730;(x + 7)] = 4 - 4x - x - 7 (I think I'm supposed to switch the sign, because I've subtracted it and moved it to the opposite side, right?)\r<BR>\n" );
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document.write( "I think I'm correct up to this point, but now I have to square both sides again.<BR>\n" );
document.write( "I think this left hand side could be re-written as:\r<BR>\n" );
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document.write( " -2[&#8730;(x + 7) * - 2&#8730;(x)]\r<BR>\n" );
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document.write( "Is this right? I'm subtracting it from itself, a negative, which could simply multiplied by -2. Anyway, now I need to square this again, so I assume the -2 becomes a 4 and I FOIL them separately?<BR>\n" );
document.write( "FOILING the left side will get:\r<BR>\n" );
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document.write( " [&#8730;(x + 7) * - 2&#8730;(x)] * [&#8730;(x + 7) * - 2&#8730;(x)]\r<BR>\n" );
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document.write( "Which, when FOILed, looks like\r<BR>\n" );
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document.write( "(x+7) - [&#8730;(x+7) * -2&#8730;(x)] - [ - 2&#8730;(x) * &#8730;(x + 7)] + 4x\r<BR>\n" );
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document.write( "It looks exactly the same as before!! I'm just really confused by this problem, and I have a couple more like it, so I want to know if figuring this one out could help me solve the other ones.\r<BR>\n" );
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document.write( "I'm confused about FOILing the different sides, whether or not I can combine two square roots, and quite frankly, a lot of other things.<BR>\n" );
document.write( "One of the options on the test is 9, and I think this is the answer, because I've inserted it into the original equation and it works, but I'm just confused about how to actually get 9 out of this..\r<BR>\n" );
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document.write( "Sorry for the long question. I hope it's not hard to understand. I'm just hoping someone can walk me through all the steps of solving a problem like this so I can do it easily in the future. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854436 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13367)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854436\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Your original equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2B7%29-2sqrt%28x%29=-2 ALIGN=MIDDLE  ALT=\"sqrt%28x%2B7%29-2sqrt%28x%29=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "You CAN start the problem by squaring both sides of the equation; but when you do that you end up with an equation involving an \"uglier\" square root.  Reaching the final answer is almost always easier if you start by changing the equation so that there is only one radical on each side of the equation.<br><BR>\n" );
document.write( "And in this particular example, with the \"-2\" on the right side, I would also, to avoid possible future confusion, move that term to the left hand side, making the equation<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2B7%29%2B2=2sqrt%28x%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2B7%29%2B2=2sqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now square both sides.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B7%29%2B4sqrt%28x%2B7%29%2B4=4x ALIGN=MIDDLE  ALT=\"%28x%2B7%29%2B4sqrt%28x%2B7%29%2B4=4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now isolate the remaining radical and square both sides again.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4sqrt%28x%2B7%29=3x-11 ALIGN=MIDDLE  ALT=\"4sqrt%28x%2B7%29=3x-11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%28x%2B7%29=9x%5E2-66x%2B121 ALIGN=MIDDLE  ALT=\"16%28x%2B7%29=9x%5E2-66x%2B121\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%2B112=9x%5E2-66x%2B121 ALIGN=MIDDLE  ALT=\"16x%2B112=9x%5E2-66x%2B121\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2-82x%2B9=0 ALIGN=MIDDLE  ALT=\"9x%5E2-82x%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289x-1%29%28x-9%29=0 ALIGN=MIDDLE  ALT=\"%289x-1%29%28x-9%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F9 ALIGN=MIDDLE  ALT=\"x=1%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9 ALIGN=MIDDLE  ALT=\"x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We squared both sides of the equation in solving the problem, so some of the roots we ended up with might not satisfy the original equation, so we need t check.  x = 1/9 does NOT satisfy the original equation; x = 9 does.  So the unique solution to the given equation is x = 9.<br><BR>\n" );
document.write( "ANSWER: x = 9<br><BR>\n" );
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document.write( "A final note....<br><BR>\n" );
document.write( "I work with high school math students who often take timed competitive tests where the speed of getting the answer is important.  While the intent here is almost certainly to solve the problem by formal mathematical methods, the problem can be solved informally VERY quickly using logical reasoning.<br><BR>\n" );
document.write( "The original equation shows that a combination via addition or subtraction of the radicals sqrt(x+7) and sqrt(x) yields in an integer result.  That means (x+7) and (x) must both be integer perfect squares.  A little knowledge of perfect square integers immediately tells us that the only two perfect square integers that differ by 7 are 16 and 9.<br><BR>\n" );
document.write( "So we quickly have our solution:<BR>\n" );
document.write( "ANSWER: x = 9<br><BR>\n" );
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