Solve 1/x+1+1/2sqrt{x+1}-54/left(x+6right)^2=y

$$\frac{ 1 }{ x+1 } + \frac{ 1 }{ 2 \sqrt{ x+1 } } - \frac{ 54 }{ { \left(x+6 \right) }^{ 2 } } =y$$

Solve for y (complex solution)

$$y=\frac{\left(x+1\right)^{-\frac{3}{2}}\left(x^{3}+2\sqrt{x+1}x^{2}+13x^{2}-84\sqrt{x+1}x+48x-36\sqrt{x+1}+36\right)}{2\left(x+6\right)^{2}}$$

$$x\neq -1\text{ and }x\neq -6$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+1\right)\left(x+6\right)^{2}, the least common multiple of x+1,\left(x+6\right)^{2}.

$$\left(x+6\right)^{2}+\frac{1}{2}\left(x+6\right)^{2}\left(x+1\right)^{\frac{1}{2}}-\left(x+1\right)\times 54=y\left(x+1\right)\left(x+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$x^{2}+12x+36+\frac{1}{2}\left(x+6\right)^{2}\left(x+1\right)^{\frac{1}{2}}-\left(x+1\right)\times 54=y\left(x+1\right)\left(x+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$x^{2}+12x+36+\frac{1}{2}\left(x^{2}+12x+36\right)\left(x+1\right)^{\frac{1}{2}}-\left(x+1\right)\times 54=y\left(x+1\right)\left(x+6\right)^{2}$$

Use the distributive property to multiply \frac{1}{2} by x^{2}+12x+36.

$$x^{2}+12x+36+\left(\frac{1}{2}x^{2}+6x+18\right)\left(x+1\right)^{\frac{1}{2}}-\left(x+1\right)\times 54=y\left(x+1\right)\left(x+6\right)^{2}$$

Use the distributive property to multiply \frac{1}{2}x^{2}+6x+18 by \left(x+1\right)^{\frac{1}{2}}.

$$x^{2}+12x+36+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}-\left(x+1\right)\times 54=y\left(x+1\right)\left(x+6\right)^{2}$$

Use the distributive property to multiply x+1 by 54.

$$x^{2}+12x+36+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}-\left(54x+54\right)=y\left(x+1\right)\left(x+6\right)^{2}$$

To find the opposite of 54x+54, find the opposite of each term.

$$x^{2}+12x+36+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}-54x-54=y\left(x+1\right)\left(x+6\right)^{2}$$

Combine 12x and -54x to get -42x.

$$x^{2}-42x+36+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}-54=y\left(x+1\right)\left(x+6\right)^{2}$$

Subtract 54 from 36 to get -18.

$$x^{2}-42x-18+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}=y\left(x+1\right)\left(x+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$x^{2}-42x-18+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}=y\left(x+1\right)\left(x^{2}+12x+36\right)$$

Use the distributive property to multiply y by x+1.

$$x^{2}-42x-18+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}=\left(yx+y\right)\left(x^{2}+12x+36\right)$$

Use the distributive property to multiply yx+y by x^{2}+12x+36 and combine like terms.

$$x^{2}-42x-18+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}=yx^{3}+13yx^{2}+48yx+36y$$

Swap sides so that all variable terms are on the left hand side.

$$yx^{3}+13yx^{2}+48yx+36y=x^{2}-42x-18+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}$$

Reorder the terms.

$$48xy+yx^{3}+13yx^{2}+36y=\frac{1}{2}\sqrt{x+1}x^{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18$$

Combine all terms containing y.

$$\left(48x+x^{3}+13x^{2}+36\right)y=\frac{1}{2}\sqrt{x+1}x^{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18$$

The equation is in standard form.

$$\left(x^{3}+13x^{2}+48x+36\right)y=\frac{\sqrt{x+1}x^{2}}{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18$$

Divide both sides by 48x+x^{3}+13x^{2}+36.

$$\frac{\left(x^{3}+13x^{2}+48x+36\right)y}{x^{3}+13x^{2}+48x+36}=\frac{\frac{\sqrt{x+1}x^{2}}{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18}{x^{3}+13x^{2}+48x+36}$$

Dividing by 48x+x^{3}+13x^{2}+36 undoes the multiplication by 48x+x^{3}+13x^{2}+36.

$$y=\frac{\frac{\sqrt{x+1}x^{2}}{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18}{x^{3}+13x^{2}+48x+36}$$

Divide \frac{\sqrt{x+1}x^{2}}{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18 by 48x+x^{3}+13x^{2}+36.

$$y=\frac{\sqrt{x+1}x^{2}+2x^{2}+12\sqrt{x+1}x-84x+36\sqrt{x+1}-36}{2\left(x+1\right)\left(x+6\right)^{2}}$$

Solve for y

$$y=\frac{\sqrt{x+1}x^{2}+2x^{2}+12\sqrt{x+1}x-84x+36\sqrt{x+1}-36}{2\left(x+1\right)\left(x+6\right)^{2}}$$

$$x>-1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+1\right)\left(x+6\right)^{2}, the least common multiple of x+1,\left(x+6\right)^{2}.

$$\left(x+6\right)^{2}+\frac{1}{2}\left(x+6\right)^{2}\left(x+1\right)^{\frac{1}{2}}-\left(x+1\right)\times 54=y\left(x+1\right)\left(x+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$x^{2}+12x+36+\frac{1}{2}\left(x+6\right)^{2}\left(x+1\right)^{\frac{1}{2}}-\left(x+1\right)\times 54=y\left(x+1\right)\left(x+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$x^{2}+12x+36+\frac{1}{2}\left(x^{2}+12x+36\right)\left(x+1\right)^{\frac{1}{2}}-\left(x+1\right)\times 54=y\left(x+1\right)\left(x+6\right)^{2}$$

Use the distributive property to multiply \frac{1}{2} by x^{2}+12x+36.

$$x^{2}+12x+36+\left(\frac{1}{2}x^{2}+6x+18\right)\left(x+1\right)^{\frac{1}{2}}-\left(x+1\right)\times 54=y\left(x+1\right)\left(x+6\right)^{2}$$

Use the distributive property to multiply \frac{1}{2}x^{2}+6x+18 by \left(x+1\right)^{\frac{1}{2}}.

$$x^{2}+12x+36+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}-\left(x+1\right)\times 54=y\left(x+1\right)\left(x+6\right)^{2}$$

Use the distributive property to multiply x+1 by 54.

$$x^{2}+12x+36+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}-\left(54x+54\right)=y\left(x+1\right)\left(x+6\right)^{2}$$

To find the opposite of 54x+54, find the opposite of each term.

$$x^{2}+12x+36+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}-54x-54=y\left(x+1\right)\left(x+6\right)^{2}$$

Combine 12x and -54x to get -42x.

$$x^{2}-42x+36+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}-54=y\left(x+1\right)\left(x+6\right)^{2}$$

Subtract 54 from 36 to get -18.

$$x^{2}-42x-18+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}=y\left(x+1\right)\left(x+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$x^{2}-42x-18+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}=y\left(x+1\right)\left(x^{2}+12x+36\right)$$

Use the distributive property to multiply y by x+1.

$$x^{2}-42x-18+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}=\left(yx+y\right)\left(x^{2}+12x+36\right)$$

Use the distributive property to multiply yx+y by x^{2}+12x+36 and combine like terms.

$$x^{2}-42x-18+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}=yx^{3}+13yx^{2}+48yx+36y$$

Swap sides so that all variable terms are on the left hand side.

$$yx^{3}+13yx^{2}+48yx+36y=x^{2}-42x-18+\frac{1}{2}x^{2}\left(x+1\right)^{\frac{1}{2}}+6x\left(x+1\right)^{\frac{1}{2}}+18\left(x+1\right)^{\frac{1}{2}}$$

Reorder the terms.

$$48xy+yx^{3}+13yx^{2}+36y=\frac{1}{2}\sqrt{x+1}x^{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18$$

Combine all terms containing y.

$$\left(48x+x^{3}+13x^{2}+36\right)y=\frac{1}{2}\sqrt{x+1}x^{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18$$

The equation is in standard form.

$$\left(x^{3}+13x^{2}+48x+36\right)y=\frac{\sqrt{x+1}x^{2}}{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18$$

Divide both sides by 48x+x^{3}+13x^{2}+36.

$$\frac{\left(x^{3}+13x^{2}+48x+36\right)y}{x^{3}+13x^{2}+48x+36}=\frac{\frac{\sqrt{x+1}x^{2}}{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18}{x^{3}+13x^{2}+48x+36}$$

Dividing by 48x+x^{3}+13x^{2}+36 undoes the multiplication by 48x+x^{3}+13x^{2}+36.

$$y=\frac{\frac{\sqrt{x+1}x^{2}}{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18}{x^{3}+13x^{2}+48x+36}$$

Divide \frac{\sqrt{x+1}x^{2}}{2}+x^{2}+6\sqrt{x+1}x-42x+18\sqrt{x+1}-18 by 48x+x^{3}+13x^{2}+36.

$$y=\frac{\sqrt{x+1}x^{2}+2x^{2}+12\sqrt{x+1}x-84x+36\sqrt{x+1}-36}{2\left(x+1\right)\left(x+6\right)^{2}}$$