Solve (x^{2}+1/x^{2})^2-5(x^2-1/x^2)^2+9/4

$$( x ^ { 2 } + \frac { 1 } { x ^ { 2 } } ) ^ { 2 } - 5 ( x ^ { 2 } - \frac { 1 } { x ^ { 2 } } ) ^ { 2 } + \frac { 9 } { 4 }$$

Evaluate

$$-4x^{4}+\frac{57}{4}-\frac{4}{x^{4}}$$

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}}.

$$\left(\frac{x^{2}x^{2}}{x^{2}}+\frac{1}{x^{2}}\right)^{2}-5\left(x^{2}-\frac{1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\left(\frac{x^{2}x^{2}+1}{x^{2}}\right)^{2}-5\left(x^{2}-\frac{1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\left(\frac{x^{4}+1}{x^{2}}\right)^{2}-5\left(x^{2}-\frac{1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-5\left(x^{2}-\frac{1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-5\left(\frac{x^{2}x^{2}}{x^{2}}-\frac{1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-5\times \left(\frac{x^{2}x^{2}-1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-5\times \left(\frac{x^{4}-1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-5\times \frac{\left(x^{4}-1\right)^{2}}{\left(x^{2}\right)^{2}}+\frac{9}{4}$$

Express 5\times \frac{\left(x^{4}-1\right)^{2}}{\left(x^{2}\right)^{2}} as a single fraction.

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-\frac{5\left(x^{4}-1\right)^{2}}{\left(x^{2}\right)^{2}}+\frac{9}{4}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-\frac{5\left(x^{4}-1\right)^{2}}{x^{4}}+\frac{9}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Expand \left(x^{2}\right)^{2}.

$$\frac{\left(x^{4}+1\right)^{2}}{x^{4}}-\frac{5\left(x^{4}-1\right)^{2}}{x^{4}}+\frac{9}{4}$$

Since \frac{\left(x^{4}+1\right)^{2}}{x^{4}} and \frac{5\left(x^{4}-1\right)^{2}}{x^{4}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{4}+1\right)^{2}-5\left(x^{4}-1\right)^{2}}{x^{4}}+\frac{9}{4}$$

Do the multiplications in \left(x^{4}+1\right)^{2}-5\left(x^{4}-1\right)^{2}.

$$\frac{x^{8}+2x^{4}+1-5x^{8}+10x^{4}-5}{x^{4}}+\frac{9}{4}$$

Combine like terms in x^{8}+2x^{4}+1-5x^{8}+10x^{4}-5.

$$\frac{-4-4x^{8}+12x^{4}}{x^{4}}+\frac{9}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{4} and 4 is 4x^{4}. Multiply \frac{-4-4x^{8}+12x^{4}}{x^{4}} times \frac{4}{4}. Multiply \frac{9}{4} times \frac{x^{4}}{x^{4}}.

$$\frac{4\left(-4-4x^{8}+12x^{4}\right)}{4x^{4}}+\frac{9x^{4}}{4x^{4}}$$

Since \frac{4\left(-4-4x^{8}+12x^{4}\right)}{4x^{4}} and \frac{9x^{4}}{4x^{4}} have the same denominator, add them by adding their numerators.

$$\frac{4\left(-4-4x^{8}+12x^{4}\right)+9x^{4}}{4x^{4}}$$

Do the multiplications in 4\left(-4-4x^{8}+12x^{4}\right)+9x^{4}.

$$\frac{-16-16x^{8}+48x^{4}+9x^{4}}{4x^{4}}$$

Combine like terms in -16-16x^{8}+48x^{4}+9x^{4}.

$$\frac{-16x^{8}-16+57x^{4}}{4x^{4}}$$

Factor the expressions that are not already factored in \frac{-16x^{8}-16+57x^{4}}{4x^{4}}.

$$\frac{-16\left(x^{4}-\left(-\frac{5}{32}\sqrt{89}+\frac{57}{32}\right)\right)\left(x^{4}-\left(\frac{5}{32}\sqrt{89}+\frac{57}{32}\right)\right)}{4x^{4}}$$

Cancel out 4 in both numerator and denominator.

$$\frac{-4\left(x^{4}-\left(-\frac{5}{32}\sqrt{89}+\frac{57}{32}\right)\right)\left(x^{4}-\left(\frac{5}{32}\sqrt{89}+\frac{57}{32}\right)\right)}{x^{4}}$$

To find the opposite of -\frac{5}{32}\sqrt{89}+\frac{57}{32}, find the opposite of each term.

$$\frac{-4\left(x^{4}+\frac{5}{32}\sqrt{89}-\frac{57}{32}\right)\left(x^{4}-\left(\frac{5}{32}\sqrt{89}+\frac{57}{32}\right)\right)}{x^{4}}$$

To find the opposite of \frac{5}{32}\sqrt{89}+\frac{57}{32}, find the opposite of each term.

$$\frac{-4\left(x^{4}+\frac{5}{32}\sqrt{89}-\frac{57}{32}\right)\left(x^{4}-\frac{5}{32}\sqrt{89}-\frac{57}{32}\right)}{x^{4}}$$

Use the distributive property to multiply -4 by x^{4}+\frac{5}{32}\sqrt{89}-\frac{57}{32}.

$$\frac{\left(-4x^{4}-\frac{5}{8}\sqrt{89}+\frac{57}{8}\right)\left(x^{4}-\frac{5}{32}\sqrt{89}-\frac{57}{32}\right)}{x^{4}}$$

Use the distributive property to multiply -4x^{4}-\frac{5}{8}\sqrt{89}+\frac{57}{8} by x^{4}-\frac{5}{32}\sqrt{89}-\frac{57}{32} and combine like terms.

$$\frac{-4x^{8}+\frac{57}{4}x^{4}+\frac{25}{256}\left(\sqrt{89}\right)^{2}-\frac{3249}{256}}{x^{4}}$$

The square of \sqrt{89} is 89.

$$\frac{-4x^{8}+\frac{57}{4}x^{4}+\frac{25}{256}\times 89-\frac{3249}{256}}{x^{4}}$$

Multiply \frac{25}{256} and 89 to get \frac{2225}{256}.

$$\frac{-4x^{8}+\frac{57}{4}x^{4}+\frac{2225}{256}-\frac{3249}{256}}{x^{4}}$$

Subtract \frac{3249}{256} from \frac{2225}{256} to get -4.

$$\frac{-4x^{8}+\frac{57}{4}x^{4}-4}{x^{4}}$$

Expand

$$-4x^{4}+\frac{57}{4}-\frac{4}{x^{4}}$$

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}}.

$$\left(\frac{x^{2}x^{2}}{x^{2}}+\frac{1}{x^{2}}\right)^{2}-5\left(x^{2}-\frac{1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\left(\frac{x^{2}x^{2}+1}{x^{2}}\right)^{2}-5\left(x^{2}-\frac{1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\left(\frac{x^{4}+1}{x^{2}}\right)^{2}-5\left(x^{2}-\frac{1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-5\left(x^{2}-\frac{1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-5\left(\frac{x^{2}x^{2}}{x^{2}}-\frac{1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-5\times \left(\frac{x^{2}x^{2}-1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-5\times \left(\frac{x^{4}-1}{x^{2}}\right)^{2}+\frac{9}{4}$$

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

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-5\times \frac{\left(x^{4}-1\right)^{2}}{\left(x^{2}\right)^{2}}+\frac{9}{4}$$

Express 5\times \frac{\left(x^{4}-1\right)^{2}}{\left(x^{2}\right)^{2}} as a single fraction.

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-\frac{5\left(x^{4}-1\right)^{2}}{\left(x^{2}\right)^{2}}+\frac{9}{4}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(x^{4}+1\right)^{2}}{\left(x^{2}\right)^{2}}-\frac{5\left(x^{4}-1\right)^{2}}{x^{4}}+\frac{9}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Expand \left(x^{2}\right)^{2}.

$$\frac{\left(x^{4}+1\right)^{2}}{x^{4}}-\frac{5\left(x^{4}-1\right)^{2}}{x^{4}}+\frac{9}{4}$$

Since \frac{\left(x^{4}+1\right)^{2}}{x^{4}} and \frac{5\left(x^{4}-1\right)^{2}}{x^{4}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{4}+1\right)^{2}-5\left(x^{4}-1\right)^{2}}{x^{4}}+\frac{9}{4}$$

Do the multiplications in \left(x^{4}+1\right)^{2}-5\left(x^{4}-1\right)^{2}.

$$\frac{x^{8}+2x^{4}+1-5x^{8}+10x^{4}-5}{x^{4}}+\frac{9}{4}$$

Combine like terms in x^{8}+2x^{4}+1-5x^{8}+10x^{4}-5.

$$\frac{-4-4x^{8}+12x^{4}}{x^{4}}+\frac{9}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{4} and 4 is 4x^{4}. Multiply \frac{-4-4x^{8}+12x^{4}}{x^{4}} times \frac{4}{4}. Multiply \frac{9}{4} times \frac{x^{4}}{x^{4}}.

$$\frac{4\left(-4-4x^{8}+12x^{4}\right)}{4x^{4}}+\frac{9x^{4}}{4x^{4}}$$

Since \frac{4\left(-4-4x^{8}+12x^{4}\right)}{4x^{4}} and \frac{9x^{4}}{4x^{4}} have the same denominator, add them by adding their numerators.

$$\frac{4\left(-4-4x^{8}+12x^{4}\right)+9x^{4}}{4x^{4}}$$

Do the multiplications in 4\left(-4-4x^{8}+12x^{4}\right)+9x^{4}.

$$\frac{-16-16x^{8}+48x^{4}+9x^{4}}{4x^{4}}$$

Combine like terms in -16-16x^{8}+48x^{4}+9x^{4}.

$$\frac{-16x^{8}-16+57x^{4}}{4x^{4}}$$

Factor the expressions that are not already factored in \frac{-16x^{8}-16+57x^{4}}{4x^{4}}.

$$\frac{-16\left(x^{4}-\left(-\frac{5}{32}\sqrt{89}+\frac{57}{32}\right)\right)\left(x^{4}-\left(\frac{5}{32}\sqrt{89}+\frac{57}{32}\right)\right)}{4x^{4}}$$

Cancel out 4 in both numerator and denominator.

$$\frac{-4\left(x^{4}-\left(-\frac{5}{32}\sqrt{89}+\frac{57}{32}\right)\right)\left(x^{4}-\left(\frac{5}{32}\sqrt{89}+\frac{57}{32}\right)\right)}{x^{4}}$$

To find the opposite of -\frac{5}{32}\sqrt{89}+\frac{57}{32}, find the opposite of each term.

$$\frac{-4\left(x^{4}+\frac{5}{32}\sqrt{89}-\frac{57}{32}\right)\left(x^{4}-\left(\frac{5}{32}\sqrt{89}+\frac{57}{32}\right)\right)}{x^{4}}$$

To find the opposite of \frac{5}{32}\sqrt{89}+\frac{57}{32}, find the opposite of each term.

$$\frac{-4\left(x^{4}+\frac{5}{32}\sqrt{89}-\frac{57}{32}\right)\left(x^{4}-\frac{5}{32}\sqrt{89}-\frac{57}{32}\right)}{x^{4}}$$

Use the distributive property to multiply -4 by x^{4}+\frac{5}{32}\sqrt{89}-\frac{57}{32}.

$$\frac{\left(-4x^{4}-\frac{5}{8}\sqrt{89}+\frac{57}{8}\right)\left(x^{4}-\frac{5}{32}\sqrt{89}-\frac{57}{32}\right)}{x^{4}}$$

Use the distributive property to multiply -4x^{4}-\frac{5}{8}\sqrt{89}+\frac{57}{8} by x^{4}-\frac{5}{32}\sqrt{89}-\frac{57}{32} and combine like terms.

$$\frac{-4x^{8}+\frac{57}{4}x^{4}+\frac{25}{256}\left(\sqrt{89}\right)^{2}-\frac{3249}{256}}{x^{4}}$$

The square of \sqrt{89} is 89.

$$\frac{-4x^{8}+\frac{57}{4}x^{4}+\frac{25}{256}\times 89-\frac{3249}{256}}{x^{4}}$$

Multiply \frac{25}{256} and 89 to get \frac{2225}{256}.

$$\frac{-4x^{8}+\frac{57}{4}x^{4}+\frac{2225}{256}-\frac{3249}{256}}{x^{4}}$$

Subtract \frac{3249}{256} from \frac{2225}{256} to get -4.

$$\frac{-4x^{8}+\frac{57}{4}x^{4}-4}{x^{4}}$$