document.write( "Question 1095268: how much xsquare -y square ? if x+y=66 xy=9
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Algebra.Com's Answer #854197 by KMST(5344)\"\" \"About 
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ONE WAY:
\n" ); document.write( "Maybe we know about solving quadratic equations.
\n" ); document.write( "Sometimes we can solve a quadratic equation such as \"z%5E2%2Bbz%2Bc=0\" by factoring if we find two integers \"p\" \"q\" such that \"p%2Bq=-b\" and \"pq=c\" .
\n" ); document.write( "Then those values for \"p\" and \"q\" are the solutions of the equation and the equation is really
\n" ); document.write( "\"z%5E2-%28p%2Bq%29z%2Bpq=%28z-p%29%28z-q%29=0\" .
\n" ); document.write( "The solutions to a quadratic equation of the form \"z%5E2%2Bbz%2Bc=0\" are always two numbers whose sum and product can be found are \"-b\" and \"c\" respectively.
\n" ); document.write( "Unfortunately, sometimes those numbers are irrational, or even imaginary, and then factoring is not an option.
\n" ); document.write( "Then, w must use algebra to \"complete the square\" and then solve, or apply the dreaded quadratic formula
\n" ); document.write( "that says the solutions to an equation of the form \"ax%5E2%2Bbx%2Bc=0\" are given by
\n" ); document.write( "\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"
\n" ); document.write( "To avoid confusion, I used \"z\" for the variable instead of \"x\" , and I prefer equations where the leading coefficient is \"a=1\" , so I wrote my equation as \"z%5E2%2Bbz%2Bc=0\" .
\n" ); document.write( "I will make \"b=-%28p%2Bq%29=-66\" and \"c=pq=9\" to get \"z%5E2-66z%2B9=0\".
\n" ); document.write( "The solutions will be the numbers \"p\" and \"q\" (or \"x\" and \"y\" ) that add up to \"x%2By=66\" and whose product is \"x%2Ay=9\" .
\n" ); document.write( "The quadratic formula tells me that the solution are given by
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\n" ); document.write( "The solutions to the equation are \"%2833%2B6sqrt%2830%29%29\" and \"%2833-12sqrt%2830%29%29\"
\n" ); document.write( "One is \"x\" and the other is \"y\" , but there is no way to guess which was intended to be \"x\" and which was intended to be \"y\" .
\n" ); document.write( "\"x%5E2\" and \"y%5E2\" are
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\n" ); document.write( "The possible answers are
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\n" ); document.write( "ANOTHER APPROACH:
\n" ); document.write( "We know that \"x%5E2-y%5E2=%28x%2By%29%28x-y%29\" . If we only knew the value of \"%28x-y%29\" we could easily find the value of \"x%5E2-y%5E2\"
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\n" ); document.write( "We know that \"%28x-y%29%5E2=x%5E2%2By%5E2-2xy\" , but to calculate \"%28x-y%29%5E2\" we would need the value of \"x%5E2%2By%5E2\"
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\n" ); document.write( "We know that \"%28x%2By%29%5E2=x%5E2%2By%5E2%2B2xy\" and we know the values of \"%28x%2By%29=66=2%2A3%2A11\" and \"xy=9=3%5E2\"
\n" ); document.write( "Substituting the known values we get \"66%5E2=x%5E2%2By%5E2%2B2%2A9\" --> \"66%5E2=x%5E2%2By%5E2%2B18\" --> \"x%5E2%2By%5E2=66%5E2-18=4338\"\r
\n" ); document.write( "\n" ); document.write( "Now we can use that value of to calculate the values of \"%28x-y%29%5E2\" and \"%28x-y%29\" , and from that find the value of \"x%5E2-y%5E2\"
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\n" ); document.write( "\"%28x-y%29%5E2=x%5E2%2By%5E2-2xy=4338-2%2A9=4338-18=4320=30%2A144=30%2A12%5E2\"
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\n" ); document.write( "Then \"%28x-y%29=%22+%22+%2B-+sqrt%2830-12%5E2%29=%22+%22+%2B-+12sqrt%2830%29\"
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\n" ); document.write( "Multiplying times \n" ); document.write( "
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