Răspuns:
Explicație pas cu pas:
a+b = 6 ; a x b = 1
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a²+b² = (a+b)²-2ab = 6²-2•1 = 36-2 = 34
a³+b³ = (a+b)³ - 3ab(a+b) = 6³-3•1•6 = 6•(36-3) = 6•33 = 198
a⁴+b⁴ = (a²+b²)²- 2a²b² = 34² - 2 = 1154
a+b = 6 ; a = 1/b =>
1/b + b = 6 => b²-6b+1 = 0 => b₁,₂ = (6±√32)/2 = 3±2√2
a = 3+2√2 ; b = 3-2√2 => a-b = 4√2
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a+b = 4 ; a³+b³ = 52
a³+b³ = (a+b)(a²-ab+b²) =>
4(a²+b²-ab) = 52 I:4 =>
a²+b²-ab = 13 =>
a²+b² = 13+ab
a³+b³ = (a+b)³-3ab(a+b) <=>
4³-3ab•4 = 52 I : 4 =>
16 - 3ab = 13 => 3ab = 3 => ab = 1 =>
a²+b² = 13+1 = 14
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a-b = 2 ; a x b = 2
(a-b)² = 4 = a²-2ab+b² = a²+b²-4 =>
a²+b² = 8
a³-b³ = (a-b)(a²+b²+ab) = 2·(8+2) = 20
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a+b = 1 ; a³+b³ = 37
a³+b³ = (a+b)(a²+b²+ab)
(a+b)² = 1 = a²+b²+2ab => a²+b² = 1-2ab =>
ab = (a²+b²-1)/2
a = 1-b =>
37 = 1·(a²+b²+ab) =>
²⁾a²+²⁾b² = ab-37 = (a²+b²-1)/2 - ²⁾37=>
2a²+2b²-a²-b²+1 = 74 =>
a²+b² = 73
