Solve frac{2(-2-sqrt{6})+4}{{left(-2-sqrt{6}right)}^2+2}

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

Evaluate

$$-\frac{\sqrt{6}}{2}+1\approx -0.224744871$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-2-\sqrt{6}\right)^{2}.

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

The square of \sqrt{6} is 6.

$$\frac{2\left(-2-\sqrt{6}\right)+4}{4+4\sqrt{6}+6+2}$$

Add 4 and 6 to get 10.

$$\frac{2\left(-2-\sqrt{6}\right)+4}{10+4\sqrt{6}+2}$$

Add 10 and 2 to get 12.

$$\frac{2\left(-2-\sqrt{6}\right)+4}{12+4\sqrt{6}}$$

Rationalize the denominator of \frac{2\left(-2-\sqrt{6}\right)+4}{12+4\sqrt{6}} by multiplying numerator and denominator by 12-4\sqrt{6}.

$$\frac{\left(2\left(-2-\sqrt{6}\right)+4\right)\left(12-4\sqrt{6}\right)}{\left(12+4\sqrt{6}\right)\left(12-4\sqrt{6}\right)}$$

Consider \left(12+4\sqrt{6}\right)\left(12-4\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2\left(-2-\sqrt{6}\right)+4\right)\left(12-4\sqrt{6}\right)}{12^{2}-\left(4\sqrt{6}\right)^{2}}$$

Calculate 12 to the power of 2 and get 144.

$$\frac{\left(2\left(-2-\sqrt{6}\right)+4\right)\left(12-4\sqrt{6}\right)}{144-\left(4\sqrt{6}\right)^{2}}$$

Expand \left(4\sqrt{6}\right)^{2}.

$$\frac{\left(2\left(-2-\sqrt{6}\right)+4\right)\left(12-4\sqrt{6}\right)}{144-4^{2}\left(\sqrt{6}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(2\left(-2-\sqrt{6}\right)+4\right)\left(12-4\sqrt{6}\right)}{144-16\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(2\left(-2-\sqrt{6}\right)+4\right)\left(12-4\sqrt{6}\right)}{144-16\times 6}$$

Multiply 16 and 6 to get 96.

$$\frac{\left(2\left(-2-\sqrt{6}\right)+4\right)\left(12-4\sqrt{6}\right)}{144-96}$$

Subtract 96 from 144 to get 48.

$$\frac{\left(2\left(-2-\sqrt{6}\right)+4\right)\left(12-4\sqrt{6}\right)}{48}$$

Use the distributive property to multiply 2 by -2-\sqrt{6}.

$$\frac{\left(-4-2\sqrt{6}+4\right)\left(12-4\sqrt{6}\right)}{48}$$

Add -4 and 4 to get 0.

$$\frac{-2\sqrt{6}\left(12-4\sqrt{6}\right)}{48}$$

Use the distributive property to multiply -2\sqrt{6} by 12-4\sqrt{6}.

$$\frac{-24\sqrt{6}+8\left(\sqrt{6}\right)^{2}}{48}$$

The square of \sqrt{6} is 6.

$$\frac{-24\sqrt{6}+8\times 6}{48}$$

Multiply 8 and 6 to get 48.

$$\frac{-24\sqrt{6}+48}{48}$$

Factor

$$\frac{2 - \sqrt{6}}{2} = -0.22474487139158894$$