Solve frac{(frac{-3+sqrt{5}}{2}-3)(frac{-3+sqrt{5}}{2})}{2}

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

Evaluate

$$-\frac{3\sqrt{5}}{2}+4\approx 0.645898034$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{2}{2}.

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

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

$$\frac{\frac{-3+\sqrt{5}-3\times 2}{2}\times \frac{-3+\sqrt{5}}{2}}{2}$$

Do the multiplications in -3+\sqrt{5}-3\times 2.

$$\frac{\frac{-3+\sqrt{5}-6}{2}\times \frac{-3+\sqrt{5}}{2}}{2}$$

Do the calculations in -3+\sqrt{5}-6.

$$\frac{\frac{-9+\sqrt{5}}{2}\times \frac{-3+\sqrt{5}}{2}}{2}$$

Multiply \frac{-9+\sqrt{5}}{2} times \frac{-3+\sqrt{5}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(-9+\sqrt{5}\right)\left(-3+\sqrt{5}\right)}{2\times 2}}{2}$$

Express \frac{\frac{\left(-9+\sqrt{5}\right)\left(-3+\sqrt{5}\right)}{2\times 2}}{2} as a single fraction.

$$\frac{\left(-9+\sqrt{5}\right)\left(-3+\sqrt{5}\right)}{2\times 2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(-9+\sqrt{5}\right)\left(-3+\sqrt{5}\right)}{4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{\left(-9+\sqrt{5}\right)\left(-3+\sqrt{5}\right)}{8}$$

Apply the distributive property by multiplying each term of -9+\sqrt{5} by each term of -3+\sqrt{5}.

$$\frac{27-9\sqrt{5}-3\sqrt{5}+\left(\sqrt{5}\right)^{2}}{8}$$

Combine -9\sqrt{5} and -3\sqrt{5} to get -12\sqrt{5}.

$$\frac{27-12\sqrt{5}+\left(\sqrt{5}\right)^{2}}{8}$$

The square of \sqrt{5} is 5.

$$\frac{27-12\sqrt{5}+5}{8}$$

Add 27 and 5 to get 32.

$$\frac{32-12\sqrt{5}}{8}$$

Factor

$$\frac{8 - 3 \sqrt{5}}{2} = 0.6458980337503153$$