Solve 5-3(n+3)/x^2+3x=1-x/x

$$5 - \frac { 3 ( n + 3 ) } { x ^ { 2 } + 3 x } = \frac { 1 - x } { x }$$

Solve for n

$$n=2x^{2}+\frac{17x}{3}-4$$

$$x\neq -3\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(x+3\right), the least common multiple of x^{2}+3x,x.

$$x\left(x+3\right)\times 5-3\left(n+3\right)=\left(x+3\right)\left(1-x\right)$$

Use the distributive property to multiply x by x+3.

$$\left(x^{2}+3x\right)\times 5-3\left(n+3\right)=\left(x+3\right)\left(1-x\right)$$

Use the distributive property to multiply x^{2}+3x by 5.

$$5x^{2}+15x-3\left(n+3\right)=\left(x+3\right)\left(1-x\right)$$

Use the distributive property to multiply -3 by n+3.

$$5x^{2}+15x-3n-9=\left(x+3\right)\left(1-x\right)$$

Use the distributive property to multiply x+3 by 1-x and combine like terms.

$$5x^{2}+15x-3n-9=-2x-x^{2}+3$$

Subtract 5x^{2} from both sides.

$$15x-3n-9=-2x-x^{2}+3-5x^{2}$$

Combine -x^{2} and -5x^{2} to get -6x^{2}.

$$15x-3n-9=-2x-6x^{2}+3$$

Subtract 15x from both sides.

$$-3n-9=-2x-6x^{2}+3-15x$$

Combine -2x and -15x to get -17x.

$$-3n-9=-17x-6x^{2}+3$$

Add 9 to both sides.

$$-3n=-17x-6x^{2}+3+9$$

Add 3 and 9 to get 12.

$$-3n=-17x-6x^{2}+12$$

The equation is in standard form.

$$-3n=12-17x-6x^{2}$$

Divide both sides by -3.

$$\frac{-3n}{-3}=\frac{12-17x-6x^{2}}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$n=\frac{12-17x-6x^{2}}{-3}$$

Divide -17x-6x^{2}+12 by -3.

$$n=2x^{2}+\frac{17x}{3}-4$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{72n+577}-17}{12}\text{, }&n\neq -3\\x=\frac{\sqrt{72n+577}-17}{12}\text{, }&n\neq -4\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{72n+577}-17}{12}\text{, }&n\neq -3\text{ and }n\geq -\frac{577}{72}\\x=\frac{\sqrt{72n+577}-17}{12}\text{, }&n\geq -\frac{577}{72}\text{ and }n\neq -4\end{matrix}\right.$$