Solve frac{-{left(frac{sqrt{5}}{2}-1right)}^2}{{left(frac{sqrt{5}}{2}right)}^2-1}

$$\frac{ - { \left( \frac{ \sqrt{ 5 } }{ 2 } -1 \right) }^{ 2 } }{ { \left( \frac{ \sqrt{ 5 } }{ 2 } \right) }^{ 2 } -1 }$$

Evaluate

$$4\sqrt{5}-9\approx -0.05572809$$

Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

$$\frac{-\left(\frac{\sqrt{5}}{2}-\frac{2}{2}\right)^{2}}{\left(\frac{\sqrt{5}}{2}\right)^{2}-1}$$

Since \frac{\sqrt{5}}{2} and \frac{2}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(\frac{\sqrt{5}-2}{2}\right)^{2}}{\left(\frac{\sqrt{5}}{2}\right)^{2}-1}$$

To raise \frac{\sqrt{5}-2}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{-\frac{\left(\sqrt{5}-2\right)^{2}}{2^{2}}}{\left(\frac{\sqrt{5}}{2}\right)^{2}-1}$$

To raise \frac{\sqrt{5}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{-\frac{\left(\sqrt{5}-2\right)^{2}}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}}{2^{2}}-1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2^{2}}{2^{2}}.

$$\frac{-\frac{\left(\sqrt{5}-2\right)^{2}}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}}{2^{2}}-\frac{2^{2}}{2^{2}}}$$

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

$$\frac{-\frac{\left(\sqrt{5}-2\right)^{2}}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}-2^{2}}{2^{2}}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-2\right)^{2}.

$$\frac{-\frac{\left(\sqrt{5}\right)^{2}-4\sqrt{5}+4}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}-2^{2}}{2^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{-\frac{5-4\sqrt{5}+4}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}-2^{2}}{2^{2}}}$$

Add 5 and 4 to get 9.

$$\frac{-\frac{9-4\sqrt{5}}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}-2^{2}}{2^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{-\frac{9-4\sqrt{5}}{4}}{\frac{\left(\sqrt{5}\right)^{2}-2^{2}}{2^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{-\frac{9-4\sqrt{5}}{4}}{\frac{5-2^{2}}{2^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{-\frac{9-4\sqrt{5}}{4}}{\frac{5-4}{2^{2}}}$$

Subtract 4 from 5 to get 1.

$$\frac{-\frac{9-4\sqrt{5}}{4}}{\frac{1}{2^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{-\frac{9-4\sqrt{5}}{4}}{\frac{1}{4}}$$

Divide -\frac{9-4\sqrt{5}}{4} by \frac{1}{4} by multiplying -\frac{9-4\sqrt{5}}{4} by the reciprocal of \frac{1}{4}.

$$\left(-\frac{9-4\sqrt{5}}{4}\right)\times 4$$

Cancel out 4 and 4.

$$-\left(9-4\sqrt{5}\right)$$

To find the opposite of 9-4\sqrt{5}, find the opposite of each term.

$$-9+4\sqrt{5}$$

Expand

$$4 \sqrt{5} - 9 = -0.05572809$$

Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

$$\frac{-\left(\frac{\sqrt{5}}{2}-\frac{2}{2}\right)^{2}}{\left(\frac{\sqrt{5}}{2}\right)^{2}-1}$$

Since \frac{\sqrt{5}}{2} and \frac{2}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(\frac{\sqrt{5}-2}{2}\right)^{2}}{\left(\frac{\sqrt{5}}{2}\right)^{2}-1}$$

To raise \frac{\sqrt{5}-2}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{-\frac{\left(\sqrt{5}-2\right)^{2}}{2^{2}}}{\left(\frac{\sqrt{5}}{2}\right)^{2}-1}$$

To raise \frac{\sqrt{5}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{-\frac{\left(\sqrt{5}-2\right)^{2}}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}}{2^{2}}-1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2^{2}}{2^{2}}.

$$\frac{-\frac{\left(\sqrt{5}-2\right)^{2}}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}}{2^{2}}-\frac{2^{2}}{2^{2}}}$$

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

$$\frac{-\frac{\left(\sqrt{5}-2\right)^{2}}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}-2^{2}}{2^{2}}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-2\right)^{2}.

$$\frac{-\frac{\left(\sqrt{5}\right)^{2}-4\sqrt{5}+4}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}-2^{2}}{2^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{-\frac{5-4\sqrt{5}+4}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}-2^{2}}{2^{2}}}$$

Add 5 and 4 to get 9.

$$\frac{-\frac{9-4\sqrt{5}}{2^{2}}}{\frac{\left(\sqrt{5}\right)^{2}-2^{2}}{2^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{-\frac{9-4\sqrt{5}}{4}}{\frac{\left(\sqrt{5}\right)^{2}-2^{2}}{2^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{-\frac{9-4\sqrt{5}}{4}}{\frac{5-2^{2}}{2^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{-\frac{9-4\sqrt{5}}{4}}{\frac{5-4}{2^{2}}}$$

Subtract 4 from 5 to get 1.

$$\frac{-\frac{9-4\sqrt{5}}{4}}{\frac{1}{2^{2}}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{-\frac{9-4\sqrt{5}}{4}}{\frac{1}{4}}$$

Divide -\frac{9-4\sqrt{5}}{4} by \frac{1}{4} by multiplying -\frac{9-4\sqrt{5}}{4} by the reciprocal of \frac{1}{4}.

$$\left(-\frac{9-4\sqrt{5}}{4}\right)\times 4$$

Cancel out 4 and 4.

$$-\left(9-4\sqrt{5}\right)$$

To find the opposite of 9-4\sqrt{5}, find the opposite of each term.

$$-9+4\sqrt{5}$$