Solve P:P=(2+x/2-x+frac{4x^{2}}{x^2-4}-2-x/2+x):x^2-3x/2x^2-x^3

$$P : P = ( \frac { 2 + x } { 2 - x } + \frac { 4 x ^ { 2 } } { x ^ { 2 } - 4 } - \frac { 2 - x } { 2 + x } ) : \frac { x ^ { 2 } - 3 x } { 2 x ^ { 2 } - x ^ { 3 } }$$

Solve for P

$$P\neq 0$$

$$x = \frac{\sqrt[3]{6 \sqrt{80229} + 1765} + \sqrt[3]{1765 - 6 \sqrt{80229}} + 7}{12} = 2.1802301552804595$$

Steps for Solving Linear Equation

Variable P cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by P.

$$P=Px\left(-3+x\right)^{-1}\left(2-x\right)\left(\frac{2+x}{2-x}+\frac{4x^{2}}{x^{2}-4}-\frac{2-x}{2+x}\right)$$

Factor x^{2}-4.

$$P=Px\left(-3+x\right)^{-1}\left(2-x\right)\left(\frac{2+x}{2-x}+\frac{4x^{2}}{\left(x-2\right)\left(x+2\right)}-\frac{2-x}{2+x}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2-x and \left(x-2\right)\left(x+2\right) is \left(x-2\right)\left(x+2\right). Multiply \frac{2+x}{2-x} times \frac{-\left(x+2\right)}{-\left(x+2\right)}.

$$P=Px\left(-3+x\right)^{-1}\left(2-x\right)\left(\frac{\left(2+x\right)\left(-1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\frac{4x^{2}}{\left(x-2\right)\left(x+2\right)}-\frac{2-x}{2+x}\right)$$

Since \frac{\left(2+x\right)\left(-1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)} and \frac{4x^{2}}{\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$P=Px\left(-3+x\right)^{-1}\left(2-x\right)\left(\frac{\left(2+x\right)\left(-1\right)\left(x+2\right)+4x^{2}}{\left(x-2\right)\left(x+2\right)}-\frac{2-x}{2+x}\right)$$

Do the multiplications in \left(2+x\right)\left(-1\right)\left(x+2\right)+4x^{2}.

$$P=Px\left(-3+x\right)^{-1}\left(2-x\right)\left(\frac{-2x-4-x^{2}-2x+4x^{2}}{\left(x-2\right)\left(x+2\right)}-\frac{2-x}{2+x}\right)$$

Combine like terms in -2x-4-x^{2}-2x+4x^{2}.

$$P=Px\left(-3+x\right)^{-1}\left(2-x\right)\left(\frac{-4x-4+3x^{2}}{\left(x-2\right)\left(x+2\right)}-\frac{2-x}{2+x}\right)$$

Factor the expressions that are not already factored in \frac{-4x-4+3x^{2}}{\left(x-2\right)\left(x+2\right)}.

$$P=Px\left(-3+x\right)^{-1}\left(2-x\right)\left(\frac{\left(x-2\right)\left(3x+2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{2-x}{2+x}\right)$$

Cancel out x-2 in both numerator and denominator.

$$P=Px\left(-3+x\right)^{-1}\left(2-x\right)\left(\frac{3x+2}{x+2}-\frac{2-x}{2+x}\right)$$

Since \frac{3x+2}{x+2} and \frac{2-x}{2+x} have the same denominator, subtract them by subtracting their numerators.

$$P=Px\left(-3+x\right)^{-1}\left(2-x\right)\times \frac{3x+2-\left(2-x\right)}{x+2}$$

Do the multiplications in 3x+2-\left(2-x\right).

$$P=Px\left(-3+x\right)^{-1}\left(2-x\right)\times \frac{3x+2-2+x}{x+2}$$

Combine like terms in 3x+2-2+x.

$$P=Px\left(-3+x\right)^{-1}\left(2-x\right)\times \frac{4x}{x+2}$$

Express P\times \frac{4x}{x+2} as a single fraction.

$$P=\frac{P\times 4x}{x+2}x\left(-3+x\right)^{-1}\left(2-x\right)$$

Use the distributive property to multiply \frac{P\times 4x}{x+2}x\left(-3+x\right)^{-1} by 2-x.

$$P=2\times \frac{P\times 4x}{x+2}x\left(-3+x\right)^{-1}-\frac{4Px}{x+2}\left(-3+x\right)^{-1}x^{2}$$

Express 2\times \frac{P\times 4x}{x+2} as a single fraction.

$$P=\frac{2P\times 4x}{x+2}x\left(-3+x\right)^{-1}-\frac{4Px}{x+2}\left(-3+x\right)^{-1}x^{2}$$

Express \frac{2P\times 4x}{x+2}x as a single fraction.

$$P=\frac{2P\times 4xx}{x+2}\left(-3+x\right)^{-1}-\frac{4Px}{x+2}\left(-3+x\right)^{-1}x^{2}$$

Express \frac{2P\times 4xx}{x+2}\left(-3+x\right)^{-1} as a single fraction.

$$P=\frac{2P\times 4xx\left(-3+x\right)^{-1}}{x+2}-\frac{4Px}{x+2}\left(-3+x\right)^{-1}x^{2}$$

Express \frac{4Px}{x+2}\left(-3+x\right)^{-1} as a single fraction.

$$P=\frac{2P\times 4xx\left(-3+x\right)^{-1}}{x+2}-\frac{4Px\left(-3+x\right)^{-1}}{x+2}x^{2}$$

Express \frac{4Px\left(-3+x\right)^{-1}}{x+2}x^{2} as a single fraction.

$$P=\frac{2P\times 4xx\left(-3+x\right)^{-1}}{x+2}-\frac{4Px\left(-3+x\right)^{-1}x^{2}}{x+2}$$

Since \frac{2P\times 4xx\left(-3+x\right)^{-1}}{x+2} and \frac{4Px\left(-3+x\right)^{-1}x^{2}}{x+2} have the same denominator, subtract them by subtracting their numerators.

$$P=\frac{2P\times 4xx\left(-3+x\right)^{-1}-4Px\left(-3+x\right)^{-1}x^{2}}{x+2}$$

Multiply x and x to get x^{2}.

$$P=\frac{2P\times 4x^{2}\left(-3+x\right)^{-1}-4Px\left(-3+x\right)^{-1}x^{2}}{x+2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$P=\frac{2P\times 4x^{2}\left(-3+x\right)^{-1}-4Px^{3}\left(-3+x\right)^{-1}}{x+2}$$

Multiply 2 and 4 to get 8.

$$P=\frac{8Px^{2}\left(-3+x\right)^{-1}-4Px^{3}\left(-3+x\right)^{-1}}{x+2}$$

Subtract \frac{8Px^{2}\left(-3+x\right)^{-1}-4Px^{3}\left(-3+x\right)^{-1}}{x+2} from both sides.

$$P-\frac{8Px^{2}\left(-3+x\right)^{-1}-4Px^{3}\left(-3+x\right)^{-1}}{x+2}=0$$

Multiply both sides of the equation by x+2.

$$\left(x+2\right)P-\left(8Px^{2}\left(-3+x\right)^{-1}-4Px^{3}\left(-3+x\right)^{-1}\right)=0$$

Reorder the terms.

$$-\left(-4\times \frac{1}{x-3}Px^{3}+8\times \frac{1}{x-3}Px^{2}\right)+P\left(x+2\right)=0$$

Multiply both sides of the equation by x-3.

$$-\left(-4\times \frac{1}{x-3}Px^{3}+8\times \frac{1}{x-3}Px^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

Express -4\times \frac{1}{x-3} as a single fraction.

$$-\left(\frac{-4}{x-3}Px^{3}+8\times \frac{1}{x-3}Px^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

Express \frac{-4}{x-3}P as a single fraction.

$$-\left(\frac{-4P}{x-3}x^{3}+8\times \frac{1}{x-3}Px^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

Express \frac{-4P}{x-3}x^{3} as a single fraction.

$$-\left(\frac{-4Px^{3}}{x-3}+8\times \frac{1}{x-3}Px^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

Express 8\times \frac{1}{x-3} as a single fraction.

$$-\left(\frac{-4Px^{3}}{x-3}+\frac{8}{x-3}Px^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

Express \frac{8}{x-3}P as a single fraction.

$$-\left(\frac{-4Px^{3}}{x-3}+\frac{8P}{x-3}x^{2}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

Express \frac{8P}{x-3}x^{2} as a single fraction.

$$-\left(\frac{-4Px^{3}}{x-3}+\frac{8Px^{2}}{x-3}\right)\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

Since \frac{-4Px^{3}}{x-3} and \frac{8Px^{2}}{x-3} have the same denominator, add them by adding their numerators.

$$-\frac{-4Px^{3}+8Px^{2}}{x-3}\left(x-3\right)+P\left(x+2\right)\left(x-3\right)=0$$

Express \frac{-4Px^{3}+8Px^{2}}{x-3}\left(x-3\right) as a single fraction.

$$-\frac{\left(-4Px^{3}+8Px^{2}\right)\left(x-3\right)}{x-3}+P\left(x+2\right)\left(x-3\right)=0$$

Cancel out x-3 in both numerator and denominator.

$$-\left(-4Px^{3}+8Px^{2}\right)+P\left(x+2\right)\left(x-3\right)=0$$

To find the opposite of -4Px^{3}+8Px^{2}, find the opposite of each term.

$$4Px^{3}-8Px^{2}+P\left(x+2\right)\left(x-3\right)=0$$

Use the distributive property to multiply P by x+2.

$$4Px^{3}-8Px^{2}+\left(Px+2P\right)\left(x-3\right)=0$$

Use the distributive property to multiply Px+2P by x-3 and combine like terms.

$$4Px^{3}-8Px^{2}+Px^{2}-Px-6P=0$$

Combine -8Px^{2} and Px^{2} to get -7Px^{2}.

$$4Px^{3}-7Px^{2}-Px-6P=0$$

Combine all terms containing P.

$$\left(4x^{3}-7x^{2}-x-6\right)P=0$$

Divide 0 by -x-7x^{2}-6+4x^{3}.

$$P=0$$

Variable P cannot be equal to 0.

$$P\in \emptyset$$

Solve for x

$$x = \frac{\sqrt[3]{6 \sqrt{80229} + 1765} + \sqrt[3]{1765 - 6 \sqrt{80229}} + 7}{12} = 2.1802301552804595$$

$$P\neq 0$$