Solve =1/4*frac{1+sqrt{5}}{4}{3-(frac{1+sqrt{5}}{4})^2}

$$= \frac { 1 } { 4 } \times \frac { 1 + \sqrt { 5 } } { 4 } \{ 3 - ( \frac { 1 + \sqrt { 5 } } { 4 } ) ^ { 2 } \}$$

Evaluate

$$\frac{5\sqrt{5}+4}{32}\approx 0.474385621$$

Short Solution Steps

To raise \frac{1+\sqrt{5}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1}{4}\times \frac{1+\sqrt{5}}{4}\left(3-\frac{\left(1+\sqrt{5}\right)^{2}}{4^{2}}\right)$$

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

$$\frac{1}{4}\times \frac{1+\sqrt{5}}{4}\left(3-\frac{1+2\sqrt{5}+\left(\sqrt{5}\right)^{2}}{4^{2}}\right)$$

The square of \sqrt{5} is 5.

$$\frac{1}{4}\times \frac{1+\sqrt{5}}{4}\left(3-\frac{1+2\sqrt{5}+5}{4^{2}}\right)$$

Add 1 and 5 to get 6.

$$\frac{1}{4}\times \frac{1+\sqrt{5}}{4}\left(3-\frac{6+2\sqrt{5}}{4^{2}}\right)$$

Calculate 4 to the power of 2 and get 16.

$$\frac{1}{4}\times \frac{1+\sqrt{5}}{4}\left(3-\frac{6+2\sqrt{5}}{16}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{16}{16}.

$$\frac{1}{4}\times \frac{1+\sqrt{5}}{4}\left(\frac{3\times 16}{16}-\frac{6+2\sqrt{5}}{16}\right)$$

Since \frac{3\times 16}{16} and \frac{6+2\sqrt{5}}{16} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{4}\times \frac{1+\sqrt{5}}{4}\times \frac{3\times 16-\left(6+2\sqrt{5}\right)}{16}$$

Do the multiplications in 3\times 16-\left(6+2\sqrt{5}\right).

$$\frac{1}{4}\times \frac{1+\sqrt{5}}{4}\times \frac{48-6-2\sqrt{5}}{16}$$

Do the calculations in 48-6-2\sqrt{5}.

$$\frac{1}{4}\times \frac{1+\sqrt{5}}{4}\times \frac{42-2\sqrt{5}}{16}$$

Multiply \frac{1}{4} times \frac{1+\sqrt{5}}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1+\sqrt{5}}{4\times 4}\times \frac{42-2\sqrt{5}}{16}$$

Multiply \frac{1+\sqrt{5}}{4\times 4} times \frac{42-2\sqrt{5}}{16} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(1+\sqrt{5}\right)\left(42-2\sqrt{5}\right)}{4\times 4\times 16}$$

Multiply 4 and 4 to get 16.

$$\frac{\left(1+\sqrt{5}\right)\left(42-2\sqrt{5}\right)}{16\times 16}$$

Multiply 16 and 16 to get 256.

$$\frac{\left(1+\sqrt{5}\right)\left(42-2\sqrt{5}\right)}{256}$$

Use the distributive property to multiply 1+\sqrt{5} by 42-2\sqrt{5} and combine like terms.

$$\frac{42+40\sqrt{5}-2\left(\sqrt{5}\right)^{2}}{256}$$

The square of \sqrt{5} is 5.

$$\frac{42+40\sqrt{5}-2\times 5}{256}$$

Multiply -2 and 5 to get -10.

$$\frac{42+40\sqrt{5}-10}{256}$$

Subtract 10 from 42 to get 32.

$$\frac{32+40\sqrt{5}}{256}$$

Factor

$$\frac{5 \sqrt{5} + 4}{32} = 0.47438562148434216$$