Solve 2{left(frac{1/xright)}^2+3{left(1/xright)}^-4}{31/x+3}

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

Evaluate

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

Short Solution Steps

To raise \frac{1}{x} to a power, raise both numerator and denominator to the power and then divide.

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

Express 2\times \frac{1^{2}}{x^{2}} as a single fraction.

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

Calculate \frac{1}{x} to the power of 1 and get \frac{1}{x}.

$$\frac{\frac{2\times 1^{2}}{x^{2}}+3\times \frac{1}{x}-4}{3\times \frac{1}{x}+3}$$

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

$$\frac{\frac{2\times 1^{2}}{x^{2}}+\frac{3}{x}-4}{3\times \frac{1}{x}+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and x is x^{2}. Multiply \frac{3}{x} times \frac{x}{x}.

$$\frac{\frac{2\times 1^{2}}{x^{2}}+\frac{3x}{x^{2}}-4}{3\times \frac{1}{x}+3}$$

Since \frac{2\times 1^{2}}{x^{2}} and \frac{3x}{x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2\times 1^{2}+3x}{x^{2}}-4}{3\times \frac{1}{x}+3}$$

Do the multiplications in 2\times 1^{2}+3x.

$$\frac{\frac{2+3x}{x^{2}}-4}{3\times \frac{1}{x}+3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{x^{2}}{x^{2}}.

$$\frac{\frac{2+3x}{x^{2}}-\frac{4x^{2}}{x^{2}}}{3\times \frac{1}{x}+3}$$

Since \frac{2+3x}{x^{2}} and \frac{4x^{2}}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2+3x-4x^{2}}{x^{2}}}{3\times \frac{1}{x}+3}$$

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x}{x}.

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

Since \frac{3}{x} and \frac{3x}{x} have the same denominator, add them by adding their numerators.

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

Divide \frac{2+3x-4x^{2}}{x^{2}} by \frac{3+3x}{x} by multiplying \frac{2+3x-4x^{2}}{x^{2}} by the reciprocal of \frac{3+3x}{x}.

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

Cancel out x in both numerator and denominator.

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

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

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

Expand

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

Short Solution Steps

To raise \frac{1}{x} to a power, raise both numerator and denominator to the power and then divide.

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

Express 2\times \frac{1^{2}}{x^{2}} as a single fraction.

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

Calculate \frac{1}{x} to the power of 1 and get \frac{1}{x}.

$$\frac{\frac{2\times 1^{2}}{x^{2}}+3\times \frac{1}{x}-4}{3\times \frac{1}{x}+3}$$

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

$$\frac{\frac{2\times 1^{2}}{x^{2}}+\frac{3}{x}-4}{3\times \frac{1}{x}+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2} and x is x^{2}. Multiply \frac{3}{x} times \frac{x}{x}.

$$\frac{\frac{2\times 1^{2}}{x^{2}}+\frac{3x}{x^{2}}-4}{3\times \frac{1}{x}+3}$$

Since \frac{2\times 1^{2}}{x^{2}} and \frac{3x}{x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2\times 1^{2}+3x}{x^{2}}-4}{3\times \frac{1}{x}+3}$$

Do the multiplications in 2\times 1^{2}+3x.

$$\frac{\frac{2+3x}{x^{2}}-4}{3\times \frac{1}{x}+3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{x^{2}}{x^{2}}.

$$\frac{\frac{2+3x}{x^{2}}-\frac{4x^{2}}{x^{2}}}{3\times \frac{1}{x}+3}$$

Since \frac{2+3x}{x^{2}} and \frac{4x^{2}}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2+3x-4x^{2}}{x^{2}}}{3\times \frac{1}{x}+3}$$

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x}{x}.

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

Since \frac{3}{x} and \frac{3x}{x} have the same denominator, add them by adding their numerators.

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

Divide \frac{2+3x-4x^{2}}{x^{2}} by \frac{3+3x}{x} by multiplying \frac{2+3x-4x^{2}}{x^{2}} by the reciprocal of \frac{3+3x}{x}.

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

Cancel out x in both numerator and denominator.

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

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

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