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

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

Evaluate

$$\frac{\sqrt{15}}{2}+\sqrt{5}-\sqrt{3}-2\approx 0.440508843$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

Add 4 and 5 to get 9.

$$\frac{-4-2\sqrt{3}}{1-\left(9+4\sqrt{5}\right)}$$

To find the opposite of 9+4\sqrt{5}, find the opposite of each term.

$$\frac{-4-2\sqrt{3}}{1-9-4\sqrt{5}}$$

Subtract 9 from 1 to get -8.

$$\frac{-4-2\sqrt{3}}{-8-4\sqrt{5}}$$

Rationalize the denominator of \frac{-4-2\sqrt{3}}{-8-4\sqrt{5}} by multiplying numerator and denominator by -8+4\sqrt{5}.

$$\frac{\left(-4-2\sqrt{3}\right)\left(-8+4\sqrt{5}\right)}{\left(-8-4\sqrt{5}\right)\left(-8+4\sqrt{5}\right)}$$

Consider \left(-8-4\sqrt{5}\right)\left(-8+4\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(-4-2\sqrt{3}\right)\left(-8+4\sqrt{5}\right)}{\left(-8\right)^{2}-\left(-4\sqrt{5}\right)^{2}}$$

Calculate -8 to the power of 2 and get 64.

$$\frac{\left(-4-2\sqrt{3}\right)\left(-8+4\sqrt{5}\right)}{64-\left(-4\sqrt{5}\right)^{2}}$$

Expand \left(-4\sqrt{5}\right)^{2}.

$$\frac{\left(-4-2\sqrt{3}\right)\left(-8+4\sqrt{5}\right)}{64-\left(-4\right)^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate -4 to the power of 2 and get 16.

$$\frac{\left(-4-2\sqrt{3}\right)\left(-8+4\sqrt{5}\right)}{64-16\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(-4-2\sqrt{3}\right)\left(-8+4\sqrt{5}\right)}{64-16\times 5}$$

Multiply 16 and 5 to get 80.

$$\frac{\left(-4-2\sqrt{3}\right)\left(-8+4\sqrt{5}\right)}{64-80}$$

Subtract 80 from 64 to get -16.

$$\frac{\left(-4-2\sqrt{3}\right)\left(-8+4\sqrt{5}\right)}{-16}$$

Use the distributive property to multiply -4-2\sqrt{3} by -8+4\sqrt{5}.

$$\frac{32-16\sqrt{5}+16\sqrt{3}-8\sqrt{3}\sqrt{5}}{-16}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{32-16\sqrt{5}+16\sqrt{3}-8\sqrt{15}}{-16}$$

Factor

$$\frac{\sqrt{15} + 2 \sqrt{5} - 2 \sqrt{3} - 4}{2} = 0.4405088430346207$$