Solve 5nx^2-5x-nx-1/x^2-1=2n+1

$$\frac { 5 n x ^ { 2 } - 5 x - n x - 1 } { x ^ { 2 } - 1 } = 2 n + 1$$

Solve for n (complex solution)

$$n=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

$$x\neq \frac{1+\sqrt{23}i}{6}\text{ and }x\neq \frac{-\sqrt{23}i+1}{6}\text{ and }x\neq -1\text{ and }x\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right).

$$5nx^{2}-5x-nx-1=2n\left(x-1\right)\left(x+1\right)+\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 2n by x-1.

$$5nx^{2}-5x-nx-1=\left(2nx-2n\right)\left(x+1\right)+\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 2nx-2n by x+1 and combine like terms.

$$5nx^{2}-5x-nx-1=2nx^{2}-2n+\left(x-1\right)\left(x+1\right)$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$5nx^{2}-5x-nx-1=2nx^{2}-2n+x^{2}-1$$

Subtract 2nx^{2} from both sides.

$$5nx^{2}-5x-nx-1-2nx^{2}=-2n+x^{2}-1$$

Combine 5nx^{2} and -2nx^{2} to get 3nx^{2}.

$$3nx^{2}-5x-nx-1=-2n+x^{2}-1$$

Add 2n to both sides.

$$3nx^{2}-5x-nx-1+2n=x^{2}-1$$

Add 5x to both sides.

$$3nx^{2}-nx-1+2n=x^{2}-1+5x$$

Add 1 to both sides.

$$3nx^{2}-nx+2n=x^{2}-1+5x+1$$

Add -1 and 1 to get 0.

$$3nx^{2}-nx+2n=x^{2}+5x$$

Combine all terms containing n.

$$\left(3x^{2}-x+2\right)n=x^{2}+5x$$

Divide both sides by 3x^{2}-x+2.

$$\frac{\left(3x^{2}-x+2\right)n}{3x^{2}-x+2}=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

Dividing by 3x^{2}-x+2 undoes the multiplication by 3x^{2}-x+2.

$$n=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

Solve for n

$$n=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

$$|x|\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right).

$$5nx^{2}-5x-nx-1=2n\left(x-1\right)\left(x+1\right)+\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 2n by x-1.

$$5nx^{2}-5x-nx-1=\left(2nx-2n\right)\left(x+1\right)+\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 2nx-2n by x+1 and combine like terms.

$$5nx^{2}-5x-nx-1=2nx^{2}-2n+\left(x-1\right)\left(x+1\right)$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$5nx^{2}-5x-nx-1=2nx^{2}-2n+x^{2}-1$$

Subtract 2nx^{2} from both sides.

$$5nx^{2}-5x-nx-1-2nx^{2}=-2n+x^{2}-1$$

Combine 5nx^{2} and -2nx^{2} to get 3nx^{2}.

$$3nx^{2}-5x-nx-1=-2n+x^{2}-1$$

Add 2n to both sides.

$$3nx^{2}-5x-nx-1+2n=x^{2}-1$$

Add 5x to both sides.

$$3nx^{2}-nx-1+2n=x^{2}-1+5x$$

Add 1 to both sides.

$$3nx^{2}-nx+2n=x^{2}-1+5x+1$$

Add -1 and 1 to get 0.

$$3nx^{2}-nx+2n=x^{2}+5x$$

Combine all terms containing n.

$$\left(3x^{2}-x+2\right)n=x^{2}+5x$$

Divide both sides by 3x^{2}-x+2.

$$\frac{\left(3x^{2}-x+2\right)n}{3x^{2}-x+2}=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

Dividing by 3x^{2}-x+2 undoes the multiplication by 3x^{2}-x+2.

$$n=\frac{x\left(x+5\right)}{3x^{2}-x+2}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{25+18n-23n^{2}}+n+5}{2\left(3n-1\right)}\text{, }&n\neq \frac{1}{3}\\x=\frac{\sqrt{25+18n-23n^{2}}+n+5}{2\left(3n-1\right)}\text{, }&n\neq -\frac{2}{3}\text{ and }n\neq \frac{1}{3}\text{ and }n\neq \frac{3}{2}\\x=\frac{1}{8}\text{, }&n=\frac{1}{3}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{25+18n-23n^{2}}+n+5}{2\left(3n-1\right)}\text{, }&n\neq \frac{1}{3}\text{ and }n\geq \frac{9-4\sqrt{41}}{23}\text{ and }n\leq \frac{4\sqrt{41}+9}{23}\\x=\frac{\sqrt{25+18n-23n^{2}}+n+5}{2\left(3n-1\right)}\text{, }&n\neq \frac{3}{2}\text{ and }n\neq \frac{1}{3}\text{ and }n\geq \frac{9-4\sqrt{41}}{23}\text{ and }n\leq \frac{4\sqrt{41}+9}{23}\text{ and }n\neq -\frac{2}{3}\\x=\frac{1}{8}\text{, }&n=\frac{1}{3}\end{matrix}\right.$$