document.write( "Question 1171432: #1. Log2(3x-7)+log2(x+2)=log2(x+1)\r
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Algebra.Com's Answer #851974 by ikleyn(52776) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " The solution to equation #1 in the post by @CPhill, giving the answer x = 3, \r
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document.write( "To check, it is enough to substitute x= 3 into equation #1.\r\n" );
document.write( "You will get then in the left side\r\n" );
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document.write( "    log_2_(3*3-7) + log_2_(3+2) = log_2_(2) + log_2_(5) = log_2_(2*5) = log_2_(10);\r\n" );
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document.write( "in the right side \r\n" );
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document.write( "    log_2_(3+1) = log_2_(4),\r\n" );
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document.write( "and even by unarmed eyes, you see that the left side is not equal to the right side.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Contradiction which ruins the solution by @CPhill into dust.\r
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\n" ); document.write( "\n" ); document.write( " Below is my correct solution.\r
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document.write( "Equation  log_2_(3x-7) + log_2_(x+2) = log_2_(x+1)  in its domain implies\r\n" );
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document.write( "    (3x-7)*(x+2) = x+1\r\n" );
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document.write( "    3x^2 - x - 14 = x+1\r\n" );
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document.write( "    3x^2 - 2x - 15 = 0.\r\n" );
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document.write( "The discriminant is  b^2 - 4ac = (-2)^2 - 4*3*(-15) = 4 + 180 = 184.\r\n" );
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document.write( "The discriminant is not a perfect square - so, the equation is not factorable.\r\n" );
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document.write( "Use the quadratic formula\r\n" );
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document.write( "     =  =  = .\r\n" );
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document.write( "The roots are   =  = -1.92744,\r\n" );
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document.write( "and             =  = 2.59411  (approximately).\r\n" );
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document.write( "The root  is not in the equation's domain - so, we reject it.\r\n" );
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document.write( "The root   is in the domain, so we accept it, and this root is the unique solution to equation #1.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Equation #1 is solved.\r
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\n" ); document.write( "\n" ); document.write( "You can check it on your own that my solution x = 2.59411 is correct, by substituting it into the equation.\r
\n" ); document.write( "\n" ); document.write( "I did it and obtained a perfect match.\r
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