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15 Sep 2014, 21:05
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tushain
TehMoUsE
Both of you incorrectly assume that the variables are integers only: 11 = u^2 - v^2 has infinitely many non-integer solutions for u and v.
30 Mar 2015, 08:54
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It seems that the only exact method to find u and v is solving for quadratic (4th power) equation, but it's also too time consuming. If someone knows quick method of finding u and v without assuming that tose numbers are integers please share it.[/quote]
If \(x = u^2 - v^2\), \(y = 2uv\) and \(z = u^2 + v^2\), and if \(x = 11\), what is the value of z?
Given: \(u^2 - v^2=11\) and \(y = 2uv\).
Question: \(u^2 + v^2=?\)
(1) y = 60 --> \(2uv=60\) --> \(4u^2v^2=3,600\).
Square \(u^2 - v^2=11\): \(u^4-2u^2v^2+v^2=121\);
Add \(4u^2v^2\) to both sides: \(u^4+2u^2v^2+v^4=121+3,600\);
Apply \(a^2+2ab+b^2=(a+b)^2\): \((u^2+v^2)^2=121+3,600\).
\(u^2+v^2=\sqrt{121+3,600}=61\).
Sufficient.
(2) u = 6 --> \(36 - v^2=11\) --> \(v^2=25\) --> \(u^2+v^2=36+25=61\). Sufficient.
Answer: D.
Hope it helps.[/quote]
Hi Bunuel
Stmtn 1
Can we use Differences of Square approach
Property - The difference between squares grows as the squares themselves get larger
Hence given u^2 - v^2 = x = 11
Thus the only numbers that fit this equation are when u = 6 and v = 5. Hence we can find Z - Sufficient
Similalry Stmtn 2 = Suff
Hence D
Pls mention if this approach is correct
Thanks
If \(x = u^2 - v^2\), \(y = 2uv\) and \(z = u^2 + v^2\), and if \(x = 11\), what is the value of z?
Given: \(u^2 - v^2=11\) and \(y = 2uv\).
Question: \(u^2 + v^2=?\)
(1) y = 60 --> \(2uv=60\) --> \(4u^2v^2=3,600\).
Square \(u^2 - v^2=11\): \(u^4-2u^2v^2+v^2=121\);
Add \(4u^2v^2\) to both sides: \(u^4+2u^2v^2+v^4=121+3,600\);
Apply \(a^2+2ab+b^2=(a+b)^2\): \((u^2+v^2)^2=121+3,600\).
\(u^2+v^2=\sqrt{121+3,600}=61\).
Sufficient.
(2) u = 6 --> \(36 - v^2=11\) --> \(v^2=25\) --> \(u^2+v^2=36+25=61\). Sufficient.
Answer: D.
Hope it helps.[/quote]
Hi Bunuel
Stmtn 1
Can we use Differences of Square approach
Property - The difference between squares grows as the squares themselves get larger
Hence given u^2 - v^2 = x = 11
Thus the only numbers that fit this equation are when u = 6 and v = 5. Hence we can find Z - Sufficient
Similalry Stmtn 2 = Suff
Hence D
Pls mention if this approach is correct
Thanks
10 Feb 2016, 23:49
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vogelleblanc
Responding to a pm:
Quote:
Note that \(x = u^2 - v^2\) and x is given as 11.
So,
u^2 - v^2 = 11
\((u + v)*(u - v) = 11\)
So \([(u + v)*(u - v)]^2 = 11^2\)
\((u+v)^2 * (u - v)^2 = 11^2\)
Does this help?
