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Popular Álgebra >

(x-ab)/(a+b)+(x-ac)/(a+c)+(x-bc)/(b+c)=a+b+c

Solución

\frac{x - ab}{a + b} + \frac{x - a c}{a + c} + \frac{x - b c}{b + c} = a + b + c

Solución

x = \frac{a ^{3} b + a ^{3} c + 3 a ^{2} b ^{2} + 3 a ^{2} c ^{2} + 7 a ^{2} b c + a b ^{3} + a c ^{3} + 7 ab c ^{2} + 7 a b ^{2} c + b c ^{3} + 3 b ^{2} c ^{2} + b ^{3} c}{a ^{2} + b ^{2} + 3 ab + c ^{2} + 3 a c + 3 b c}; (a + b) (a + c) (b + c) \neq = 0, a \neq = \frac{- 3 b - 3 c + 5 b ^{2} + 6 b c + 5 c ^{2}}{2}, a \neq = \frac{- 3 b - 3 c - 5 b ^{2} + 6 b c + 5 c ^{2}}{2}

Pasos de solución

\frac{x - ab}{a + b} + \frac{x - a c}{a + c} + \frac{x - b c}{b + c} = a + b + c

Multiplicar por el mínimo común múltiplo

(x - ab) (a + c) (b + c) + (x - a c) (a + b) (b + c) + (x - b c) (a + b) (a + c) = a (a + b) (a + c) (b + c) + b (a + b) (a + c) (b + c) + c (a + b) (a + c) (b + c); (a + b) (a + c) (b + c) \neq = 0

Desarrollar

(x - ab) (a + c) (b + c) + (x - a c) (a + b) (b + c) + (x - b c) (a + b) (a + c) : a^{2} x + b^{2} x + 3 ab x + c^{2} x + 3 a c x + 3 b c x - a^{2} b^{2} - 3 ab c^{2} - a^{2} c^{2} - b^{2} c^{2} - 3 a^{2} b c - 3 a b^{2} c

Desarrollar

a (a + b) (a + c) (b + c) + b (a + b) (a + c) (b + c) + c (a + b) (a + c) (b + c) : a^{3} b + a^{3} c + 2 a^{2} b^{2} + 2 a^{2} c^{2} + 4 a^{2} b c + a b^{3} + a c^{3} + 4 ab c^{2} + 4 a b^{2} c + b c^{3} + 2 b^{2} c^{2} + b^{3} c

a^{2} x + b^{2} x + 3 ab x + c^{2} x + 3 a c x + 3 b c x - a^{2} b^{2} - 3 ab c^{2} - a^{2} c^{2} - b^{2} c^{2} - 3 a^{2} b c - 3 a b^{2} c = a^{3} b + a^{3} c + 2 a^{2} b^{2} + 2 a^{2} c^{2} + 4 a^{2} b c + a b^{3} + a c^{3} + 4 ab c^{2} + 4 a b^{2} c + b c^{3} + 2 b^{2} c^{2} + b^{3} c; (a + b) (a + c) (b + c) \neq = 0

Desplace

a^{2} b^{2}

a la derecha

a^{2} x + b^{2} x + 3 ab x + c^{2} x + 3 a c x + 3 b c x - 3 ab c^{2} - a^{2} c^{2} - b^{2} c^{2} - 3 a^{2} b c - 3 a b^{2} c = a^{3} b + a^{3} c + 3 a^{2} b^{2} + 2 a^{2} c^{2} + 4 a^{2} b c + a b^{3} + a c^{3} + 4 ab c^{2} + 4 a b^{2} c + b c^{3} + 2 b^{2} c^{2} + b^{3} c; (a + b) (a + c) (b + c) \neq = 0

Desplace

3 ab c^{2}

a la derecha

a^{2} x + b^{2} x + 3 ab x + c^{2} x + 3 a c x + 3 b c x - a^{2} c^{2} - b^{2} c^{2} - 3 a^{2} b c - 3 a b^{2} c = a^{3} b + a^{3} c + 3 a^{2} b^{2} + 2 a^{2} c^{2} + 4 a^{2} b c + a b^{3} + a c^{3} + 7 ab c^{2} + 4 a b^{2} c + b c^{3} + 2 b^{2} c^{2} + b^{3} c; (a + b) (a + c) (b + c) \neq = 0

Desplace

a^{2} c^{2}

a la derecha

a^{2} x + b^{2} x + 3 ab x + c^{2} x + 3 a c x + 3 b c x - b^{2} c^{2} - 3 a^{2} b c - 3 a b^{2} c = a^{3} b + a^{3} c + 3 a^{2} b^{2} + 3 a^{2} c^{2} + 4 a^{2} b c + a b^{3} + a c^{3} + 7 ab c^{2} + 4 a b^{2} c + b c^{3} + 2 b^{2} c^{2} + b^{3} c; (a + b) (a + c) (b + c) \neq = 0

Desplace

b^{2} c^{2}

a la derecha

a^{2} x + b^{2} x + 3 ab x + c^{2} x + 3 a c x + 3 b c x - 3 a^{2} b c - 3 a b^{2} c = a b^{3} + b^{3} c + 3 a^{2} b^{2} + 3 b^{2} c^{2} + 4 a b^{2} c + a^{3} b + b c^{3} + 7 ab c^{2} + 4 a^{2} b c + a c^{3} + 3 a^{2} c^{2} + a^{3} c; (a + b) (a + c) (b + c) \neq = 0

Desplace

3 a^{2} b c

a la derecha

a^{2} x + b^{2} x + 3 ab x + c^{2} x + 3 a c x + 3 b c x - 3 a b^{2} c = a^{3} b + a^{3} c + 3 a^{2} b^{2} + 3 a^{2} c^{2} + 7 a^{2} b c + a b^{3} + a c^{3} + 7 ab c^{2} + 4 a b^{2} c + b c^{3} + 3 b^{2} c^{2} + b^{3} c; (a + b) (a + c) (b + c) \neq = 0

Desplace

3 a b^{2} c

a la derecha

a^{2} x + b^{2} x + 3 ab x + c^{2} x + 3 a c x + 3 b c x = a^{3} b + a^{3} c + 3 a^{2} b^{2} + 3 a^{2} c^{2} + 7 a^{2} b c + a b^{3} + a c^{3} + 7 ab c^{2} + 7 a b^{2} c + b c^{3} + 3 b^{2} c^{2} + b^{3} c; (a + b) (a + c) (b + c) \neq = 0

Factorizar

a^{2} x + b^{2} x + 3 ab x + c^{2} x + 3 a c x + 3 b c x : x (a^{2} + b^{2} + 3 ab + c^{2} + 3 a c + 3 b c)

x (a^{2} + b^{2} + 3 ab + c^{2} + 3 a c + 3 b c) = a^{3} b + a^{3} c + 3 a^{2} b^{2} + 3 a^{2} c^{2} + 7 a^{2} b c + a b^{3} + a c^{3} + 7 ab c^{2} + 7 a b^{2} c + b c^{3} + 3 b^{2} c^{2} + b^{3} c; (a + b) (a + c) (b + c) \neq = 0

Dividir ambos lados entre

a^{2} + b^{2} + 3 ab + c^{2} + 3 a c + 3 b c; (a + b) (a + c) (b + c) \neq = 0, a \neq = \frac{- 3 b - 3 c + 5 b ^{2} + 6 b c + 5 c ^{2}}{2}, a \neq = \frac{- 3 b - 3 c - 5 b ^{2} + 6 b c + 5 c ^{2}}{2}

x = \frac{a ^{3} b + a ^{3} c + 3 a ^{2} b ^{2} + 3 a ^{2} c ^{2} + 7 a ^{2} b c + a b ^{3} + a c ^{3} + 7 ab c ^{2} + 7 a b ^{2} c + b c ^{3} + 3 b ^{2} c ^{2} + b ^{3} c}{a ^{2} + b ^{2} + 3 ab + c ^{2} + 3 a c + 3 b c}; (a + b) (a + c) (b + c) \neq = 0, a \neq = \frac{- 3 b - 3 c + 5 b ^{2} + 6 b c + 5 c ^{2}}{2}, a \neq = \frac{- 3 b - 3 c - 5 b ^{2} + 6 b c + 5 c ^{2}}{2}

Gráfica

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