Question

a^2=b+3, b^2=a+3

Oblicz

a+b

a^3-2ab+b^3

Answer 1

a² = b+3     => b = a²-3

b² = a+3

-------------  (-)

 

a²-b² = b-a

(a+b)(a-b) = -(a-b)  /:(a-b)                  Z: a ≠ b

a+b = -1

a+a²-3 = -1

a²+a-2 = 0

Δ = 1+8 = 9

√Δ = 3 

a1 = (-1-3)/2 = -2

a2 = (-1+3)/2 = 1

b1 = (-2)²-3 = 4-3 = 1

b2 = 1²-3 = -2

 

{a = -2}

{b = 1}

lub

{a = 1}

{b = -2}

 

dla a = -2,  b = 1

a+b = -2+1 = -1

dla a = 1,  b = -2

a+b = 1-2 = -1

 

a+b = -1

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dla a = -2, b = 1

a³-2ab+b³ = (-2)³-2·(-2)·1+1³ = -8+4+1 = -3

dla a = 1, b = (-2)

a³-2ab+b³ = 1³-2·1·(-2)+(-2)³ = 1+4-8 = -3

 

a³-2ab+b³ = -3

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Answer 2