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2x^2-6x+6=c_{1}(x^2+x-1)+c_{2}(x^2-x+1)

Solución

2 x^{2} - 6 x + 6 = c_{1} (x^{2} + x - 1) + c_{2} (x^{2} - x + 1)

x = \frac{6 - c _{2} + c _{1} + 5 c _{1}^{2} + 28 c _{1} - 2 c _{1} c _{2} + 20 c _{2} - 3 c _{2}^{2} - 12}{2 ( 2 - c _{2} - c _{1} )}, x = \frac{6 - c _{2} + c _{1} - 5 c _{1}^{2} + 28 c _{1} - 2 c _{1} c _{2} + 20 c _{2} - 3 c _{2}^{2} - 12}{2 ( 2 - c _{2} - c _{1} )}; 2 - c_{2} - c_{1} \neq = 0

Pasos de solución

2 x^{2} - 6 x + 6 = c_{1} (x^{2} + x - 1) + c_{2} (x^{2} - x + 1)

Desarrollar

c_{1} (x^{2} + x - 1) + c_{2} (x^{2} - x + 1) : c_{1} x^{2} + c_{1} x - c_{1} + c_{2} x^{2} - c_{2} x + c_{2}

2 x^{2} - 6 x + 6 = c_{1} x^{2} + c_{1} x - c_{1} + c_{2} x^{2} - c_{2} x + c_{2}

Restar

- c_{2} x + c_{2}

de ambos lados

2 x^{2} - 6 x + 6 - (- c_{2} x + c_{2}) = c_{1} x^{2} + c_{1} x - c_{1} + c_{2} x^{2} - c_{2} x + c_{2} - (- c_{2} x + c_{2})

Simplificar

2 x^{2} - 6 x + 6 + c_{2} x - c_{2} = c_{1} x^{2} + c_{1} x - c_{1} + c_{2} x^{2}

Mover

c_{2} x^{2}

al lado izquierdo

2 x^{2} - 6 x + 6 + c_{2} x - c_{2} - c_{2} x^{2} = c_{1} x^{2} + c_{1} x - c_{1}

Restar

c_{1} x - c_{1}

de ambos lados

2 x^{2} - 6 x + 6 + c_{2} x - c_{2} - c_{2} x^{2} - (c_{1} x - c_{1}) = c_{1} x^{2} + c_{1} x - c_{1} - (c_{1} x - c_{1})

Simplificar

2 x^{2} - 6 x + 6 + c_{2} x - c_{2} - c_{2} x^{2} - c_{1} x + c_{1} = c_{1} x^{2}

Mover

c_{1} x^{2}

al lado izquierdo

2 x^{2} - 6 x + 6 + c_{2} x - c_{2} - c_{2} x^{2} - c_{1} x + c_{1} - c_{1} x^{2} = 0

Escribir en la forma binómica

a x^{2} + b x + c = 0

(2 - c_{2} - c_{1}) x^{2} + (- 6 + c_{2} - c_{1}) x + 6 - c_{2} + c_{1} = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

x_{1, 2} = \frac{- ( - 6 + c _{2} - c _{1} ) \pm ( - 6 + c _{2} - c _{1} ) ^{2} - 4 ( 2 - c _{2} - c _{1} ) ( 6 - c _{2} + c _{1} )}{2 ( 2 - c _{2} - c _{1} )}

Simplificar

(- 6 + c_{2} - c_{1})^{2} - 4 (2 - c_{2} - c_{1}) (6 - c_{2} + c_{1}) : 5 c_{1}^{2} + 28 c_{1} - 2 c_{1} c_{2} + 20 c_{2} - 3 c_{2}^{2} - 12

x_{1, 2} = \frac{- ( - 6 + c _{2} - c _{1} ) \pm 5 c _{1}^{2} + 28 c _{1} - 2 c _{1} c _{2} + 20 c _{2} - 3 c _{2}^{2} - 12}{2 ( 2 - c _{2} - c _{1} )}; 2 - c_{2} - c_{1} \neq = 0

Separar las soluciones

x_{1} = \frac{- ( - 6 + c _{2} - c _{1} ) + 5 c _{1}^{2} + 28 c _{1} - 2 c _{1} c _{2} + 20 c _{2} - 3 c _{2}^{2} - 12}{2 ( 2 - c _{2} - c _{1} )}, x_{2} = \frac{- ( - 6 + c _{2} - c _{1} ) - 5 c _{1}^{2} + 28 c _{1} - 2 c _{1} c _{2} + 20 c _{2} - 3 c _{2}^{2} - 12}{2 ( 2 - c _{2} - c _{1} )}

x = \frac{- ( - 6 + c _{2} - c _{1} ) + 5 c _{1}^{2} + 28 c _{1} - 2 c _{1} c _{2} + 20 c _{2} - 3 c _{2}^{2} - 12}{2 ( 2 - c _{2} - c _{1} )} : \frac{6 - c _{2} + c _{1} + 5 c _{1}^{2} + 28 c _{1} - 2 c _{1} c _{2} + 20 c _{2} - 3 c _{2}^{2} - 12}{2 ( 2 - c _{2} - c _{1} )}

x = \frac{- ( - 6 + c _{2} - c _{1} ) - 5 c _{1}^{2} + 28 c _{1} - 2 c _{1} c _{2} + 20 c _{2} - 3 c _{2}^{2} - 12}{2 ( 2 - c _{2} - c _{1} )} : \frac{6 - c _{2} + c _{1} - 5 c _{1}^{2} + 28 c _{1} - 2 c _{1} c _{2} + 20 c _{2} - 3 c _{2}^{2} - 12}{2 ( 2 - c _{2} - c _{1} )}

Las soluciones a la ecuación de segundo grado son:

x = \frac{6 - c _{2} + c _{1} + 5 c _{1}^{2} + 28 c _{1} - 2 c _{1} c _{2} + 20 c _{2} - 3 c _{2}^{2} - 12}{2 ( 2 - c _{2} - c _{1} )}, x = \frac{6 - c _{2} + c _{1} - 5 c _{1}^{2} + 28 c _{1} - 2 c _{1} c _{2} + 20 c _{2} - 3 c _{2}^{2} - 12}{2 ( 2 - c _{2} - c _{1} )}; 2 - c_{2} - c_{1} \neq = 0

x^{2} + 11 = 2

(x - 3)^{2} = 48

(y - 1)^{2} = 3

b^{2} + 10 b + 21 = 0

7 x^{2} - 42 = 0