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(6x^2)/((sqrt(2x+1)+1)^2)=2x+sqrt(x-1)+1

Solución

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

Solución

x = 23.56067 \dots

Pasos de solución

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

Multiplicar ambos lados por

(2 x + 1 + 1)^{2}

\frac{6 x ^{2}}{( 2 x + 1 + 1 ) ^{2}} (2 x + 1 + 1)^{2} = 2 x (2 x + 1 + 1)^{2} + x - 1 (2 x + 1 + 1)^{2} + 1 \cdot (2 x + 1 + 1)^{2}

Simplificar

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

Desarrollar

2 x (2 x + 1 + 1)^{2} + x - 1 (2 x + 1 + 1)^{2} + (2 x + 1 + 1)^{2} : 4 x^{2} + 6 x + 4 x 2 x + 1 + 2 x x - 1 + 2 x - 1 + 2 x - 1 2 x + 1 + 2 2 x + 1 + 2

6 x^{2} = 4 x^{2} + 6 x + 4 x 2 x + 1 + 2 x x - 1 + 2 x - 1 + 2 x - 1 2 x + 1 + 2 2 x + 1 + 2

Re-escribir la ecuación con

x - 1 = u

y

x = u^{2} + 1

6 (u^{2} + 1)^{2} = 4 (u^{2} + 1)^{2} + 6 (u^{2} + 1) + 4 (u^{2} + 1) 2 (u^{2} + 1) + 1 + 2 (u^{2} + 1) u + 2 u + 2 u 2 (u^{2} + 1) + 1 + 2 2 (u^{2} + 1) + 1 + 2

Resolver

6 (u^{2} + 1)^{2} = 4 (u^{2} + 1)^{2} + 6 (u^{2} + 1) + 4 (u^{2} + 1) 2 (u^{2} + 1) + 1 + 2 (u^{2} + 1) u + 2 u + 2 u 2 (u^{2} + 1) + 1 + 2 2 (u^{2} + 1) + 1 + 2 : u \approx - 2.52756 \dots, u \approx 4.74980 \dots

u \approx - 2.52756 \dots, u \approx 4.74980 \dots

Sustituir hacia atrás la

u = x - 1,

resolver para

x

Resolver

x - 1 = - 2.52756 \dots :

Sin solución para

x \in R

Resolver

x - 1 = 4.74980 \dots : x = 23.56067 \dots

x = 23.56067 \dots

Verificar las soluciones:

x = 23.56067 \dots

Verdadero

La solución es

x = 23.56067 \dots

Ejemplos populares

c= 1/((sqrt(x^2+x-2)))

c = \frac{1}{( x ^{2} + x - 2 )}

2x^2+2sqrt(5x)-5=0

2 x^{2} + 2 5 x - 5 = 0

((x-1)/x)^{1/2}+((1-1)/x)^{1/2}=x

(\frac{x - 1}{x})^{\frac{1}{2}} + (\frac{1 - 1}{x})^{\frac{1}{2}} = x

sqrt(x+1)+sqrt(2x+3)=5

x + 1 + 2 x + 3 = 5

2 x - 3 = - 1