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

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

Evaluate

$$-1$$

Short Solution Steps

To raise \frac{1+\sqrt{5}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{5}\right)^{2}.

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

The square of \sqrt{5} is 5.

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

Add 1 and 5 to get 6.

$$\frac{1+\sqrt{5}}{2}-\frac{6+2\sqrt{5}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{1+\sqrt{5}}{2}-\frac{6+2\sqrt{5}}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{1+\sqrt{5}}{2} times \frac{2}{2}.

$$\frac{2\left(1+\sqrt{5}\right)}{4}-\frac{6+2\sqrt{5}}{4}$$

Since \frac{2\left(1+\sqrt{5}\right)}{4} and \frac{6+2\sqrt{5}}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(1+\sqrt{5}\right)-\left(6+2\sqrt{5}\right)}{4}$$

Do the multiplications in 2\left(1+\sqrt{5}\right)-\left(6+2\sqrt{5}\right).

$$\frac{2+2\sqrt{5}-6-2\sqrt{5}}{4}$$

Do the calculations in 2+2\sqrt{5}-6-2\sqrt{5}.

$$\frac{-4}{4}$$

Divide -4 by 4 to get -1.

$$-1$$

Factor

$$-1$$