document.write( "Question 46750: sqrt(x+3)+ sqrt(2x-3)=6 \n" ); document.write( "
Algebra.Com's Answer #30940 by AnlytcPhil(1806) $\"\"$ $\"About$
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document.write( "Öx+3 + Ö2x-3 = 6\r\n" );
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document.write( "Isolate either radical term\r\n" );
document.write( "        ____        ___ \r\n" );
document.write( "       Ö2x-3 = 6 - Öx+3\r\n" );
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document.write( "Square both sides:\r\n" );
document.write( "        ____           ___ \r\n" );
document.write( "      (Ö2x-3)² = (6 - Öx+3)²\r\n" );
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document.write( "It's easy to square the left side, for the squaring\r\n" );
document.write( "just cancels the square root.  However it's not so\r\n" );
document.write( "easy to square the right side.  Put it down twice\r\n" );
document.write( "and use FOIL\r\n" );
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document.write( "          2x-3 = (6 - Öx+3)(6 - Öx+3)\r\n" );
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document.write( "On the right:\r\n" );
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document.write( "\"F\" = 6·6 = 36\r\n" );
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document.write( "\"O\" = (6)(-Öx+3) = -6Öx+3\r\n" );
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document.write( "\"I\" = (-Öx+3)(6) = -6Öx+3\r\n" );
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document.write( "\"L\" = (-Öx+3)(-Öx+3) = (-Öx+3)² = x+3 \r\n" );
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document.write( "So we have:\r\n" );
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document.write( "         2x-3 = 36 - 6Öx+3 - 6Öx+3 + x+3\r\n" );
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document.write( "         2x-3 = 36 - 12Öx+3 + x+3\r\n" );
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document.write( "         2x-3 = 39 - 12Öx+3 + x\r\n" );
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document.write( "Isolate the radical term\r\n" );
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document.write( "       12Öx+3 = 39 + x - 2x + 3\r\n" );
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document.write( "       12Öx+3 = 42 - x \r\n" );
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document.write( "Square both sides:\r\n" );
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document.write( "     (12Öx+3)² = (42 - x)²\r\n" );
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document.write( "      144(x+3) = (42 - x)(42 - x)\r\n" );
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document.write( "    144x + 432 = 1764 - 42x - 42x + x²\r\n" );
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document.write( "    144x + 432 = 1764 - 84x + x²\r\n" );
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document.write( "Get 0 on the left:\r\n" );
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document.write( "             0 = x² - 228x + 1332\r\n" );
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document.write( "The right side may factor. However since the\r\n" );
document.write( "numbers are so big, it's probably easier to use\r\n" );
document.write( "the quadratic formula:\r\n" );
document.write( "                __________________\r\n" );
document.write( "     -(-228) ± Ö(-228)²-4(1)(1332)\r\n" );
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document.write( "                2(1)\r\n" );
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document.write( "      228 ± Ö46656\r\n" );
document.write( "x = ----------------\r\n" );
document.write( "            2\r\n" );
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document.write( "      228 ± 216\r\n" );
document.write( "x = -------------\r\n" );
document.write( "          2\r\n" );
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document.write( "Using the +\r\n" );
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document.write( "      228 + 216\r\n" );
document.write( "x = -------------\r\n" );
document.write( "          2\r\n" );
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document.write( "     444\r\n" );
document.write( "x = ----- = 222\r\n" );
document.write( "      2\r\n" );
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document.write( "Using the -\r\n" );
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document.write( "      228 - 216\r\n" );
document.write( "x = -------------\r\n" );
document.write( "          2\r\n" );
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document.write( "     12\r\n" );
document.write( "x = ---- = 6\r\n" );
document.write( "      2\r\n" );
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document.write( "Now we must check, because often we get extraneous \r\n" );
document.write( "answers in equations with even roots. Checking the\r\n" );
document.write( "answer x = 222\r\n" );
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document.write( "           ___    ____\r\n" );
document.write( "          Öx+3 + Ö2x-3 = 6\r\n" );
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document.write( "    Ö222+3 + Ö2(222)-3 = 6\r\n" );
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document.write( "           Ö225 + Ö441 = 6\r\n" );
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document.write( "               15 + 21 = 6\r\n" );
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document.write( "                    36 = 6\r\n" );
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document.write( "No that doesn't check. So we discard that answer.\r\n" );
document.write( "Checking the answer x=6\r\n" );
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document.write( "           ___    ____\r\n" );
document.write( "          Öx+3 + Ö2x-3 = 6\r\n" );
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document.write( "        Ö6+3 + Ö2(6)-3 = 6\r\n" );
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document.write( "               Ö9 + Ö9 = 6\r\n" );
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document.write( "                 3 + 3 = 6\r\n" );
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document.write( "                     6 = 6\r\n" );
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document.write( "That checks.  So there is one solution,\r\n" );
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document.write( "x = 6\r\n" );
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document.write( "Edwin
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