document.write( "Question 1208374: 1. Solve the system and graph the curves: (x + 1) ^ 2 + 2 * (y - 4) ^ 2 = 12; y ^ 2 - 8y = 4x - 16 \n" ); document.write( "

Algebra.Com's Answer #846761 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "1. Solve the system and graph the curves: (x + 1) ^ 2 + 2 * (y - 4) ^ 2 = 12; y ^ 2 - 8y = 4x - 16
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document.write( "Starting equations are\r\n" );
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document.write( "    (x+1)^2 + 2*(y-4)^2 = 12,    (1)\r\n" );
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document.write( "    y^2 - 8y = 4x - 16.          (2)\r\n" );
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document.write( "Rewrite equation (2) this way\r\n" );
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document.write( "    y^2 - 8y + 16 = 4x,\r\n" );
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document.write( "    (y-4)^2 = 4x.                (3)\r\n" );
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document.write( "Now replace the term  2*(y-4)^2  in equation (1) by  2*(4x), based on equation (3).\r\n" );
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document.write( "You will get simple quadratic equation for single unknown x\r\n" );
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document.write( "    (x+1)^2 + 8x = 12.\r\n" );
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document.write( "Write it in the standard form and solve by factoring \r\n" );
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document.write( "    (x+1)^2 + 8x - 12 = 0,\r\n" );
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document.write( "    x^2 + 2x + 1 + 8x - 12 = 0,\r\n" );
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document.write( "     x^2 + 10x - 11 = 0,\r\n" );
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document.write( "     (x+11)*(x-1) = 0.\r\n" );
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document.write( "The roots are  x= -11  and  x= 1.\r\n" );
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document.write( "For x= -11, the corresponding values of y are\r\n" );
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document.write( "    (y-4)^2 = 4*(-11)    (from equation (3)\r\n" );
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document.write( "      y-4 = +/- ,  y =    <<<---===  complex values\r\n" );
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document.write( "For x= 1, the corresponding values of y are\r\n" );
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document.write( "    (y-4)^2 = 4*1    (from equation (3)\r\n" );
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document.write( "      y-4 = +/- 2,    \r\n" );
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document.write( "which implies\r\n" );
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document.write( "      y = 4 + 2 = 6  or  y = 4 - 2 = 2.\r\n" );
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document.write( "Thus the given system (1), (2) has two real solutions  (x.y) = (1,6)   and  (x,y) = (1,2).\r\n" );
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document.write( "ANSWER.  The given system (1), (2) has two real solutions  (x.y) = (1,6)   and  (x,y) = (1,2).\r\n" );
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document.write( "         Equation (1) describes an ellipse; equation (2) describes a parabola\r\n" );
document.write( "         having the horizontal axis, and these two curves have two intersection points  (1,6)  and  (1,2).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "To make plots, go to website www.desmos.com/calculator\r
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\n" ); document.write( "\n" ); document.write( "It has free of charge plotting tool for common use.\r
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\n" ); document.write( "\n" ); document.write( "Print your equations there and get the graphs instantly.\r
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