Solve frac{sqrt{5}-1}{2}-1/sqrt{5+1/2}

$$\frac{ \sqrt{ 5 } -1 }{ 2 } - \frac{ 1 }{ \frac{ \sqrt{ 5 } +1 }{ 2 } }$$

Evaluate

$$0$$

Short Solution Steps

Divide 1 by \frac{\sqrt{5}+1}{2} by multiplying 1 by the reciprocal of \frac{\sqrt{5}+1}{2}.

$$\frac{\sqrt{5}-1}{2}-\frac{2}{\sqrt{5}+1}$$

Rationalize the denominator of \frac{2}{\sqrt{5}+1} by multiplying numerator and denominator by \sqrt{5}-1.

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

Consider \left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{5}. Square 1.

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

Subtract 1 from 5 to get 4.

$$\frac{\sqrt{5}-1}{2}-\frac{2\left(\sqrt{5}-1\right)}{4}$$

Divide 2\left(\sqrt{5}-1\right) by 4 to get \frac{1}{2}\left(\sqrt{5}-1\right).

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

Use the distributive property to multiply \frac{1}{2} by \sqrt{5}-1.

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

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

To find the opposite of \frac{1}{2}\sqrt{5}-\frac{1}{2}, find the opposite of each term.

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

The opposite of -\frac{1}{2} is \frac{1}{2}.

$$\frac{\sqrt{5}-1}{2}-\frac{1}{2}\sqrt{5}+\frac{1}{2}$$

Since \frac{\sqrt{5}-1}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{5}-1+1}{2}-\frac{1}{2}\sqrt{5}$$

Do the calculations in \sqrt{5}-1+1.

$$\frac{\sqrt{5}}{2}-\frac{1}{2}\sqrt{5}$$

Combine \frac{\sqrt{5}}{2} and -\frac{1}{2}\sqrt{5} to get 0.

$$0$$

Factor

$$0$$