Algebra Quint
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1 549
√(x²+y²+xy) - √(x²+y²-xy) = √2
So
(x²+y²+xy) + (x²+y²-xy) + 2√(x⁴+y⁴+x²y²) = 2
2(x²+y²) + 2√(x⁴+y⁴+x²y²) = 2
(x²+y²) + √(x⁴+y⁴+x²y²) = 1
√(x⁴+y⁴+x²y²) = 1 - (x²+y²)
x⁴+y⁴+x²y² = 1 + x⁴+y⁴+2x²y² - 2(x²+y²)
2(x²+y²) = x²y² + 1
2x² - x²y² + 2y² = 1
x²(2 - y²) - 4 + 2y² = 1 - 4
x²(2 - y²) - 2(2 - y²) = - 3
(2 - y²)(x² - 2) = -3
(2 - x²)(2 - y²) = 3
1 549
√(x²+y²+xy) - √(x²+y²-xy) = √2
So
(x²+y²+xy) + (x²+y²-xy) + 2√(x⁴+y⁴+x²y²) = 2
2(x²+y²) + 2√(x⁴+y⁴+x²y²) = 2
(x²+y²) + √(x⁴+y⁴+x²y²) = 1
√(x⁴+y⁴+x²y²) = 1 - (x²+y²)
x⁴+y⁴+x²y² = 1 + x⁴+y⁴+2x²y² - 2(x²+y²)
2(x²+y²) = x²y² + 1
2x² - x²y² + 2y² = 1
x²(2 - y²) - 4 + 2y² = 1 - 4
x²(2 - y²) - 2(2 - y²) = - 3
(2 - y²)(x² - 2) = -3
(2 - x²)(2 - y²) = 3
1 549
√(x²+y²+xy)-√(x²+y²-xy)=√2
xy=1+√2.√(x²+y²-xy)
x²y²+1=2x²+2y²
2x²+2y²-x²y²-4=-3
x²(2-y²)-2(2-y²)=-3
(2-x²)(2-y²)=3
1 549
(x+3)(x+4)x(x-1)+4=0
x(x+3)(x-1)(x+4)+4=0
(x²+3x)(x²+3x-4)+4=0
Put...... x²+3x=y
y(y-4)+4=0
y²-4y+4=0
y²-2y-2y+4=0
y(y-2)-2(y-2)=0
y=2
x²+3x=2
x-2/x=-3
x²+4/x²-4=9
x²+4/x²=13✅
1 549
x²-11x+1=0
x+1/x==11
x²+1/x²=119
x⁴+1/x⁴=14159
x⁸+1=14159x⁴
x⁸-14159x⁴=-1
x⁸-14159x⁴+11=-1+11=10✅
