document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=272595>Question 272595</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you help me solve the equation (2x^2-xy+y^2=8)and(xy=4). I have to find the value of x and y. I'm having trouble solving this particular problem. I just know the answer has to have 4 solutions. Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #199382 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=199382\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you have 2 equations that I presume must be solved simultaneously.\r<BR>\n" );
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document.write( "Those equations are:\r<BR>\n" );
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document.write( "(2x^2-xy+y^2=8)and(xy=4)\r<BR>\n" );
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document.write( "first thing you should be able to do is replace xy in the first equation with 4.\r<BR>\n" );
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document.write( "your first equation becomes:\r<BR>\n" );
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document.write( "2x^2 - 4 + y^2 = 8\r<BR>\n" );
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document.write( "add 4 to both sides of this equation to get:\r<BR>\n" );
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document.write( "2x^2 + y^2 = 8 + 4 which becomes:\r<BR>\n" );
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document.write( "2x^2 + y^2 = 12\r<BR>\n" );
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document.write( "since your second equation states that xy = 4, you can solve for y to get:\r<BR>\n" );
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document.write( "y = 4/x\r<BR>\n" );
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document.write( "replace y in your first equation with 4/x to get:\r<BR>\n" );
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document.write( "2x^2 + (4/x)^2 = 12\r<BR>\n" );
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document.write( "since (4/x)^2 = 4^2/x^2 = 16/x^2, your equation becomes:\r<BR>\n" );
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document.write( "2x^2 + 16/x^2 = 12\r<BR>\n" );
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document.write( "multiply both sides of this equation by x^2 to get:\r<BR>\n" );
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document.write( "2x^4 + 16 = 12x^2\r<BR>\n" );
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document.write( "subtract 12x^2 from both sides of this equation to get:\r<BR>\n" );
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document.write( "2x^4 - 12x^2 + 16 = 0\r<BR>\n" );
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document.write( "divide both sides of this equation by 2 to get:\r<BR>\n" );
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document.write( "x^4 - 6x^2 + 8 = 0\r<BR>\n" );
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document.write( "let k = x^2\r<BR>\n" );
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document.write( "your equation becomes:\r<BR>\n" );
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document.write( "k^2 - 6k + 8 = 0\r<BR>\n" );
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document.write( "factor this quadratic equation to get:\r<BR>\n" );
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document.write( "(k-2) * (k-4) = 0\r<BR>\n" );
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document.write( "since either one of these factors can be equal to 0 to make this equation true, then you have:\r<BR>\n" );
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document.write( "k-2 = 0 or k-4 = 0\r<BR>\n" );
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document.write( "solve for k to get:\r<BR>\n" );
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document.write( "k = 2 or k = 4\r<BR>\n" );
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document.write( "plugging those values into the quadratic equation will make that equation true.\r<BR>\n" );
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document.write( "the quadratic equation is k^2 - 6k + 8 = 0\r<BR>\n" );
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document.write( "when k = 2, this equation becomes 4 - 12 + 8 = 0 which is true.<BR>\n" );
document.write( "when k = 4, this equation becomes 16 - 24 + 8 = 0 which is also true.\r<BR>\n" );
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document.write( "the values for k are both good.\r<BR>\n" );
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document.write( "since k = x^2 ( you set it that way), then you can solve for x to get:\r<BR>\n" );
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document.write( "x = +/- square root of k\r<BR>\n" );
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document.write( "when k = 4, x = +/- 2\r<BR>\n" );
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document.write( "when k = 2, x = +/- square root of 2.\r<BR>\n" );
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document.write( "we also know that y = 4/x\r<BR>\n" );
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document.write( "this means that:\r<BR>\n" );
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document.write( "when x = 2, y = 4/2 = 2<BR>\n" );
document.write( "when x = -2, y = 4/(-2) = -2<BR>\n" );
document.write( "square root of 2 is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "when x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F%28sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"4%2F%28sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Asqrt%282%29%29%2F2 ALIGN=MIDDLE  ALT=\"%284%2Asqrt%282%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "when x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%282%29 ALIGN=MIDDLE  ALT=\"-sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F%28-sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"4%2F%28-sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2A%28-sqrt%282%29%29%29%2F2 ALIGN=MIDDLE  ALT=\"%284%2A%28-sqrt%282%29%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28-sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"2%2A%28-sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"-2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "plug these values into your original equations and see whether those equations become true.\r<BR>\n" );
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document.write( "your original equations are:\r<BR>\n" );
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document.write( "(2x^2-xy+y^2=8)and(xy=4)\r<BR>\n" );
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document.write( "Since xy = 4, we changed the first equation to:\r<BR>\n" );
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document.write( "2x^2 - 4 + y^2 = 8\r<BR>\n" );
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document.write( "original equation is 2x^2 - 4 + y^2 = 8\r<BR>\n" );
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document.write( "when x = 2 and y = 2, this equation becomes:\r<BR>\n" );
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document.write( "2*2^2 - 4 + 2^2 = 8 which becomes:<BR>\n" );
document.write( "2*4 - 4 + 4 = 8 which becomes:<BR>\n" );
document.write( "8 - 4 + 4 = 8 which becomes:<BR>\n" );
document.write( "8 = 8 which is true.\r<BR>\n" );
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document.write( "original equation is 2x^2 - 4 + y^2 = 8\r<BR>\n" );
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document.write( "when x = -2 and y = -2, this equation becomes:\r<BR>\n" );
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document.write( "2*(-2)^2 - 4 + (-2)^2 = 8 which becomes:<BR>\n" );
document.write( "2*4 - 4 + 4 = 8 which becomes:<BR>\n" );
document.write( "8 - 4 + 4 = 8 which becomes:<BR>\n" );
document.write( "8 = 8 which is true.\r<BR>\n" );
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document.write( "original equation is 2x^2 - 4 + y^2 = 8\r<BR>\n" );
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document.write( "when x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this equation becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28sqrt%282%29%29%5E2+-+4 ALIGN=MIDDLE  ALT=\"2%2A%28sqrt%282%29%29%5E2+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Asqrt%282%29%29%5E2 ALIGN=MIDDLE  ALT=\"%282%2Asqrt%282%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 8 which becomes:<BR>\n" );
document.write( "2*2 - 4 + 4*2 = 8 which becomes:<BR>\n" );
document.write( "4 - 4 + 8 = 8 which becomes:<BR>\n" );
document.write( "8 = 8 which is true.\r<BR>\n" );
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document.write( "original equation is 2x^2 - 4 + y^2 = 8\r<BR>\n" );
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document.write( "when x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%282%29 ALIGN=MIDDLE  ALT=\"-sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"-2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this equation becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28-sqrt%282%29%29%5E2+-+4 ALIGN=MIDDLE  ALT=\"2%2A%28-sqrt%282%29%29%5E2+-+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%2Asqrt%282%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28-2%2Asqrt%282%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 8 which becomes:<BR>\n" );
document.write( "2*2 - 4 + 4*2 = 8 which becomes:<BR>\n" );
document.write( "4 - 4 + 8 = 8 which becomes:<BR>\n" );
document.write( "8 = 8 which is true.\r<BR>\n" );
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document.write( "your second equation is xy = 4\r<BR>\n" );
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document.write( "when x = 2 and y = 2, this equation becomes 4 = 4 which is true.\r<BR>\n" );
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document.write( "when x = -2 and y = -2, this equation becomes 4 = 4 which is true.\r<BR>\n" );
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document.write( "when x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this equation becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2A2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2A2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 4 which becomes:<BR>\n" );
document.write( "2*2 = 4 which becomes:<BR>\n" );
document.write( "4 = 4 which is true.\r<BR>\n" );
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document.write( "when x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%282%29 ALIGN=MIDDLE  ALT=\"-sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"-2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this equation becomes:<BR>\n" );
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document.write( "2 * 2 = 4 which becomes<BR>\n" );
document.write( "4 = 4 which is true.\r<BR>\n" );
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document.write( "All equations are true so these values are good.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "when x = 2, y = 2<BR>\n" );
document.write( "when x = -2, y = -2<BR>\n" );
document.write( "when x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "when x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%282%29 ALIGN=MIDDLE  ALT=\"-sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, y = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"-2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "those are your 4 \"sets\" of answers.<BR>\n" );
document.write( "each answer is a pair of values, 1 for x and 1 for y.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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