document.write( "Question 162565: Solve the simultaneous equations: 32x^2+15y^2=2112
\n" ); document.write( " 7x^2-3y^2=60\r
\n" ); document.write( "\n" ); document.write( "What is the substitution for the quadratic formula for equation:
\n" ); document.write( " 8x^2+3a^2=10ax\r
\n" ); document.write( "\n" ); document.write( "What would the problem like at the beginning: x^2+y^2=65
\n" ); document.write( " 1/2xy=14\r
\n" ); document.write( "\n" ); document.write( "if discriminant of a complete quadratic equation is 8, what is the nature of its roots.\r
\n" ); document.write( "\n" ); document.write( "separate 72 into two parts so that the first part is the square of the second. let x represent the first part, what is the equation that would solve the problem. \n" ); document.write( "
Algebra.Com's Answer #119857 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Solve the simultaneous equations: \r\n" );
document.write( "\r\n" );
document.write( "\"system%2832x%5E2%2B15y%5E2=2112%2C7x%5E2-3y%5E2=60%29\"\r\n" );
document.write( "\r\n" );
document.write( "Multiply the second equation through by 5 to\r\n" );
document.write( "make the \"y%5E2\"-terms cancel when we add\r\n" );
document.write( "the equations vertically term by term:\r\n" );
document.write( "\r\n" );
document.write( "\"system%2832x%5E2%2B15y%5E2=2112%2C35x%5E2-15y%5E2=300%29\"\r\n" );
document.write( "\r\n" );
document.write( "Adding them term by term:\r\n" );
document.write( "\r\n" );
document.write( "\"67x%5E2=2412\"\r\n" );
document.write( "Divide both sides by 67\r\n" );
document.write( "\"x%5E2=2412%2F67\"\r\n" );
document.write( "\"x%5E2=36\"\r\n" );
document.write( "\"x%5E2-36=0\"\r\n" );
document.write( "Factor:\r\n" );
document.write( "\"%28x-6%29%28x%2B6%29=0\"\r\n" );
document.write( "\r\n" );
document.write( "\"matrix%282%2C3%2C+x-6=0%2C+%22%2C%22%2C+x%2B6=0%2C++x=6%2C+%22%2C%22%2C+x=-6%29\"\r\n" );
document.write( "\r\n" );
document.write( "Substitute \"x=6\" into \r\n" );
document.write( "\r\n" );
document.write( "\"7x%5E2-3y%5E2=60\"\r\n" );
document.write( "\"7%286%29%5E2-3y%5E2=60\"\r\n" );
document.write( "\"7%2836%29-3y%5E2=60\"\r\n" );
document.write( "\"252-3y%5E2=60\"\r\n" );
document.write( "\"-3y%5E2=-192\"\r\n" );
document.write( "Divide both sides by \"-3\":\r\n" );
document.write( "\r\n" );
document.write( "\"y%5E2=64\"\r\n" );
document.write( "\"y%5E2-64=0\"\r\n" );
document.write( "\"%28y-8%29%28y%2B8%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"matrix%282%2C3%2C+y-8=0%2C+%22%2C%22%2C+y%2B8=0%2C++y=8%2C+%22%2C%22%2C+y=-8%29\"\r\n" );
document.write( "\r\n" );
document.write( "So we have these solutions for \"x=6\",\r\n" );
document.write( "\r\n" );
document.write( "(x,y) = (6,8)\r\n" );
document.write( "(x,y) = (6,-8)\r\n" );
document.write( "\r\n" );
document.write( "Substitute \"x=-6\" into \r\n" );
document.write( "\r\n" );
document.write( "\"7x%5E2-3y%5E2=60\"\r\n" );
document.write( "\"7%28-6%29%5E2-3y%5E2=60\"\r\n" );
document.write( "\"7%2836%29-3y%5E2=60\"\r\n" );
document.write( "\"252-3y%5E2=60\"\r\n" );
document.write( "\"-3y%5E2=-192\"\r\n" );
document.write( "Divide both sides by \"-3\":\r\n" );
document.write( "\r\n" );
document.write( "\"y%5E2=64\"\r\n" );
document.write( "\"y%5E2-64=0\"\r\n" );
document.write( "\"%28y-8%29%28y%2B8%29=0\"\r\n" );
document.write( "\r\n" );
document.write( "\"matrix%282%2C3%2C+y-8=0%2C+%22%2C%22%2C+y%2B8=0%2C++y=8%2C+%22%2C%22%2C+y=-8%29\"\r\n" );
document.write( "\r\n" );
document.write( "So we have these solutions for \"x=-6\",\r\n" );
document.write( "\r\n" );
document.write( "(x,y) = (-6,8)\r\n" );
document.write( "(x,y) = (-6,-8)\r\n" );
document.write( "\r\n" );
document.write( "So there are four solutions:\r\n" );
document.write( "\r\n" );
document.write( "(x,y) = (6,8)\r\n" );
document.write( "(x,y) = (6,-8)\r\n" );
document.write( "(x,y) = (-6,8)\r\n" );
document.write( "(x,y) = (-6,-8)\r\n" );
document.write( "\r\n" );
document.write( "-----------------------\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "What is the substitution for the quadratic formula for equation:\r\n" );
document.write( "                                   \r\n" );
document.write( "\r\n" );
document.write( "\"8x%5E2%2B3a%5E2=10ax\"\r\n" );
document.write( "\r\n" );
document.write( "\"8x%5E2-10ax%2B3a%5E2=0\"\r\n" );
document.write( "\r\n" );
document.write( "Since this already contains a small letter \"a\", we'll\r\n" );
document.write( "write the quadratic formula using all CAPITAL LETTERS:\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%28-B+%2B-+sqrt%28+B%5E2-4%2AA%2AC+%29%29%2F%282%2AA%29+\" \r\n" );
document.write( "\r\n" );
document.write( "where \"A=8\", \"B=-10a\", and \"C=3a%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%2810a+%2B-+sqrt%28+100a%5E2-96a%5E2%29+%29%2F16+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%2810a+%2B-+sqrt%284a%5E2%29+%29%2F16+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%2810a+%2B-+2a%29%2F16+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x=2a%285+%2B-+2%29%2F16\"\r\n" );
document.write( "\r\n" );
document.write( "\"x=a%285%2B-2%29%2F8\"\r\n" );
document.write( "\r\n" );
document.write( "Using the +,\r\n" );
document.write( "\r\n" );
document.write( "\"x=a%285%2B2%29%2F8=7a%2F8\"\r\n" );
document.write( "\r\n" );
document.write( "Using the +,\r\n" );
document.write( "\r\n" );
document.write( "\"x=a%285-2%29%2F8=3a%2F8\"\r\n" );
document.write( "\r\n" );
document.write( "------------------------------ \r\n" );
document.write( "\r\n" );
document.write( "What would the problem like at the beginning:  \r\n" );
document.write( "\r\n" );
document.write( "\"system%28x%5E2%2By%5E2=65%2C+%281%2F2%29xy=14%29\"\r\n" );
document.write( "\r\n" );
document.write( "To clear of fractions, multiply second equation through\r\n" );
document.write( "by 2:\r\n" );
document.write( "\r\n" );
document.write( "\"system%28x%5E2%2By%5E2=65%2C+xy=28%29\"\r\n" );
document.write( "\r\n" );
document.write( "Solve the second equation for y:\r\n" );
document.write( "\r\n" );
document.write( "\"xy=28\"\r\n" );
document.write( "\"y=28%2Fx\"\r\n" );
document.write( "\r\n" );
document.write( "Substitute that in the first equation:\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2%2By%5E2=65\"\r\n" );
document.write( "\"x%5E2%2B%2828%2Fx%29%5E2=65\"\r\n" );
document.write( "\"x%5E2%2B784%2Fx%5E2=65\"\r\n" );
document.write( "\"x%5E4%2B784=65x%5E2\"\r\n" );
document.write( "\"x%5E4-65x%5E2%2B784=0\"\r\n" );
document.write( "\"%28x%5E2-49%29%28x%5E2-16%29=0\"\r\n" );
document.write( "\"%28x-7%29%28x%2B7%29%28x-4%29%28x%2B4%29=0\"\r\n" );
document.write( "\r\n" );
document.write( "  \r\n" );
document.write( "\r\n" );
document.write( "Substitute each of those into:\r\n" );
document.write( "\r\n" );
document.write( "\"xy=28\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So there are 4 solutions:\r\n" );
document.write( "\r\n" );
document.write( "(x,y)=(7,4)\r\n" );
document.write( "(x,y)=(-7,-4)\r\n" );
document.write( "(x,y)=(4,7)\r\n" );
document.write( "(x,y)=(-4,-7)\r\n" );
document.write( "\r\n" );
document.write( "----------------------\r\n" );
document.write( "\r\n" );
document.write( "if the discriminant of a complete quadratic equation is 8, \r\n" );
document.write( "what is the nature of its roots.\r\n" );
document.write( "\r\n" );
document.write( "When the discriminant is positive, there are two different real solutions.\r\n" );
document.write( "When it is 0 there is exactly one real solution\r\n" );
document.write( "When it is negative there are no real solutions.\r\n" );
document.write( "\r\n" );
document.write( "8 is positive so there are two different real solutions.\r\n" );
document.write( "\r\n" );
document.write( "-----------------------\r\n" );
document.write( "\r\n" );
document.write( "separate 72 into two parts so that the first part is the square of the second. let x represent the first part, what is the equation that would solve the problem.\r\n" );
document.write( "\r\n" );
document.write( "The other part of \"72\" is \"72-x\"\r\n" );
document.write( "\r\n" );
document.write( "First part: \"x\"\r\n" );
document.write( "Second part: \"72-x\"\r\n" );
document.write( "\r\n" );
document.write( ">>...the first part is the square of the second part...<<\r\n" );
document.write( "\r\n" );
document.write( "So  \r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%2872-x%29%5E2\"\r\n" );
document.write( "\"x+=+%2872-x%29%2872-x%29\"\r\n" );
document.write( "\"x+=+5184-72x-72x%2Bx%5E2\"\r\n" );
document.write( "\"x+=+5184-144x%2Bx%5E2\"\r\n" );
document.write( "\"0+=+5184-145x%2Bx%5E2\"\r\n" );
document.write( "\"0+=+x%5E2-145x%2B5184\"\r\n" );
document.write( "\"0+=+%28x-81%29%28x-64%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"matrix%282%2C3%2C+x-81=0%2C+%22%2C%22%2C+x-64=0%2C++x=81%2C+%22%2C%22%2C+x=64%29\"\r\n" );
document.write( "\r\n" );
document.write( "So if x = first part = 81, then second part = 72-81=-9\r\n" );
document.write( "\r\n" );
document.write( "That's kind of weird, separating 72 into 81 and -9, but\r\n" );
document.write( "they do sum to 72, and certainly 81 is the square of -9.\r\n" );
document.write( "\r\n" );
document.write( "Now if x = first part = 64, then second part = 72-64=8\r\n" );
document.write( "\r\n" );
document.write( "That's not weird, separating 72 into 64 and 8, and\r\n" );
document.write( "they do sum to 72, and certainly 64 is the square of 8.\r\n" );
document.write( "\r\n" );
document.write( "So there are two answers, a weird one and one not weird at all.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );