Solve -{left(frac{3-sqrt{5}}{2}right)}^2+2*frac{3-sqrt{5}}{2}+3

$$- { \left( \frac{ 3- \sqrt{ 5 } }{ 2 } \right) }^{ 2 } +2 \times \frac{ 3- \sqrt{ 5 } }{ 2 } +3$$

Evaluate

$$\frac{\sqrt{5}+5}{2}\approx 3.618033989$$

Short Solution Steps

To raise \frac{3-\sqrt{5}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$-\frac{\left(3-\sqrt{5}\right)^{2}}{2^{2}}+2\times \frac{3-\sqrt{5}}{2}+3$$

Cancel out 2 and 2.

$$-\frac{\left(3-\sqrt{5}\right)^{2}}{2^{2}}+3-\sqrt{5}+3$$

Add 3 and 3 to get 6.

$$-\frac{\left(3-\sqrt{5}\right)^{2}}{2^{2}}+6-\sqrt{5}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 6-\sqrt{5} times \frac{2^{2}}{2^{2}}.

$$-\frac{\left(3-\sqrt{5}\right)^{2}}{2^{2}}+\frac{\left(6-\sqrt{5}\right)\times 2^{2}}{2^{2}}$$

Since -\frac{\left(3-\sqrt{5}\right)^{2}}{2^{2}} and \frac{\left(6-\sqrt{5}\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-\left(3-\sqrt{5}\right)^{2}+\left(6-\sqrt{5}\right)\times 2^{2}}{2^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-\sqrt{5}\right)^{2}.

$$-\frac{9-6\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2^{2}}+6-\sqrt{5}$$

The square of \sqrt{5} is 5.

$$-\frac{9-6\sqrt{5}+5}{2^{2}}+6-\sqrt{5}$$

Add 9 and 5 to get 14.

$$-\frac{14-6\sqrt{5}}{2^{2}}+6-\sqrt{5}$$

Calculate 2 to the power of 2 and get 4.

$$-\frac{14-6\sqrt{5}}{4}+6-\sqrt{5}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 6-\sqrt{5} times \frac{4}{4}.

$$-\frac{14-6\sqrt{5}}{4}+\frac{4\left(6-\sqrt{5}\right)}{4}$$

Since -\frac{14-6\sqrt{5}}{4} and \frac{4\left(6-\sqrt{5}\right)}{4} have the same denominator, add them by adding their numerators.

$$\frac{-\left(14-6\sqrt{5}\right)+4\left(6-\sqrt{5}\right)}{4}$$

Do the multiplications in -\left(14-6\sqrt{5}\right)+4\left(6-\sqrt{5}\right).

$$\frac{-14+6\sqrt{5}+24-4\sqrt{5}}{4}$$

Do the calculations in -14+6\sqrt{5}+24-4\sqrt{5}.

$$\frac{10+2\sqrt{5}}{4}$$

Factor

$$\frac{\sqrt{5} {(\sqrt{5} + 1)}}{2} = 3.6180339887498953$$