Solve (x^2+1/x^2)^2-4(x+1/x)^2+12

$$( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } ) ^ { 2 } - 4 ( x + \frac { 1 } { x } ) ^ { 2 } + 12$$

Evaluate

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

Short Solution Steps

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

$$\left(\frac{x^{2}x^{2}}{x^{2}}+\frac{1}{x^{2}}\right)^{2}-4\left(x+\frac{1}{x}\right)^{2}+12$$

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

$$\left(\frac{x^{2}x^{2}+1}{x^{2}}\right)^{2}-4\left(x+\frac{1}{x}\right)^{2}+12$$

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

$$\left(\frac{x^{4}+1}{x^{2}}\right)^{2}-4\left(x+\frac{1}{x}\right)^{2}+12$$

To raise \frac{x^{4}+1}{x^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-4\left(x+\frac{1}{x}\right)^{2}+12$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-4\left(\frac{xx}{x}+\frac{1}{x}\right)^{2}+12$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-4\times \left(\frac{xx+1}{x}\right)^{2}+12$$

Do the multiplications in xx+1.

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-4\times \left(\frac{x^{2}+1}{x}\right)^{2}+12$$

To raise \frac{x^{2}+1}{x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-4\times \frac{\left(x^{2}+1\right)^{2}}{x^{2}}+12$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-\frac{4\left(x^{2}+1\right)^{2}}{x^{2}}+12$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x^{2}\right)^{2} and x^{2} is x^{4}. Multiply \frac{4\left(x^{2}+1\right)^{2}}{x^{2}} times \frac{x^{2}}{x^{2}}.

$$\frac{\left(x^{4}+1\right)^{2}}{x^{4}}-\frac{4\left(x^{2}+1\right)^{2}x^{2}}{x^{4}}+12$$

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

$$\frac{\left(x^{4}+1\right)^{2}-4\left(x^{2}+1\right)^{2}x^{2}}{x^{4}}+12$$

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

$$\frac{x^{8}+2x^{4}+1-4x^{6}-8x^{4}-4x^{2}}{x^{4}}+12$$

Combine like terms in x^{8}+2x^{4}+1-4x^{6}-8x^{4}-4x^{2}.

$$\frac{1+x^{8}-6x^{4}-4x^{6}-4x^{2}}{x^{4}}+12$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 12 times \frac{x^{4}}{x^{4}}.

$$\frac{1+x^{8}-6x^{4}-4x^{6}-4x^{2}}{x^{4}}+\frac{12x^{4}}{x^{4}}$$

Since \frac{1+x^{8}-6x^{4}-4x^{6}-4x^{2}}{x^{4}} and \frac{12x^{4}}{x^{4}} have the same denominator, add them by adding their numerators.

$$\frac{1+x^{8}-6x^{4}-4x^{6}-4x^{2}+12x^{4}}{x^{4}}$$

Combine like terms in 1+x^{8}-6x^{4}-4x^{6}-4x^{2}+12x^{4}.

$$\frac{x^{8}+1+6x^{4}-4x^{6}-4x^{2}}{x^{4}}$$

Expand

$$x^{4}-4x^{2}+6-\frac{4}{x^{2}}+\frac{1}{x^{4}}$$

Short Solution Steps

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

$$\left(\frac{x^{2}x^{2}}{x^{2}}+\frac{1}{x^{2}}\right)^{2}-4\left(x+\frac{1}{x}\right)^{2}+12$$

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

$$\left(\frac{x^{2}x^{2}+1}{x^{2}}\right)^{2}-4\left(x+\frac{1}{x}\right)^{2}+12$$

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

$$\left(\frac{x^{4}+1}{x^{2}}\right)^{2}-4\left(x+\frac{1}{x}\right)^{2}+12$$

To raise \frac{x^{4}+1}{x^{2}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-4\left(x+\frac{1}{x}\right)^{2}+12$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-4\left(\frac{xx}{x}+\frac{1}{x}\right)^{2}+12$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-4\times \left(\frac{xx+1}{x}\right)^{2}+12$$

Do the multiplications in xx+1.

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-4\times \left(\frac{x^{2}+1}{x}\right)^{2}+12$$

To raise \frac{x^{2}+1}{x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-4\times \frac{\left(x^{2}+1\right)^{2}}{x^{2}}+12$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-\frac{4\left(x^{2}+1\right)^{2}}{x^{2}}+12$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x^{2}\right)^{2} and x^{2} is x^{4}. Multiply \frac{4\left(x^{2}+1\right)^{2}}{x^{2}} times \frac{x^{2}}{x^{2}}.

$$\frac{\left(x^{4}+1\right)^{2}}{x^{4}}-\frac{4\left(x^{2}+1\right)^{2}x^{2}}{x^{4}}+12$$

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

$$\frac{\left(x^{4}+1\right)^{2}-4\left(x^{2}+1\right)^{2}x^{2}}{x^{4}}+12$$

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

$$\frac{x^{8}+2x^{4}+1-4x^{6}-8x^{4}-4x^{2}}{x^{4}}+12$$

Combine like terms in x^{8}+2x^{4}+1-4x^{6}-8x^{4}-4x^{2}.

$$\frac{1+x^{8}-6x^{4}-4x^{6}-4x^{2}}{x^{4}}+12$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 12 times \frac{x^{4}}{x^{4}}.

$$\frac{1+x^{8}-6x^{4}-4x^{6}-4x^{2}}{x^{4}}+\frac{12x^{4}}{x^{4}}$$

Since \frac{1+x^{8}-6x^{4}-4x^{6}-4x^{2}}{x^{4}} and \frac{12x^{4}}{x^{4}} have the same denominator, add them by adding their numerators.

$$\frac{1+x^{8}-6x^{4}-4x^{6}-4x^{2}+12x^{4}}{x^{4}}$$

Combine like terms in 1+x^{8}-6x^{4}-4x^{6}-4x^{2}+12x^{4}.

$$\frac{x^{8}+1+6x^{4}-4x^{6}-4x^{2}}{x^{4}}$$