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

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

Evaluate

$$\frac{5-\sqrt{5}}{2}\approx 1.381966011$$

Short Solution Steps

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

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

Cancel out 2 and 2.

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

Add 3 and 3 to get 6.

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

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

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

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

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

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

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

The square of \sqrt{5} is 5.

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

Add 5 and 9 to get 14.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

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

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

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

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

Factor

$$\frac{\sqrt{5} {(\sqrt{5} - 1)}}{2} = 1.3819660112501053$$