Solve frac{1-4sqrt{5}-3}{3-sqrt{5}-sqrt{5}+2}=

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

Evaluate

$$-\frac{24\sqrt{5}}{5}-10\approx -20.733126292$$

Solution Steps

Subtract 3 from 1 to get -2.

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

Combine -\sqrt{5} and -\sqrt{5} to get -2\sqrt{5}.

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

Add 3 and 2 to get 5.

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

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

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

Consider \left(5-2\sqrt{5}\right)\left(5+2\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}.

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

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(-2-4\sqrt{5}\right)\left(5+2\sqrt{5}\right)}{25-\left(-2\sqrt{5}\right)^{2}}$$

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

$$\frac{\left(-2-4\sqrt{5}\right)\left(5+2\sqrt{5}\right)}{25-\left(-2\right)^{2}\left(\sqrt{5}\right)^{2}}$$

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

$$\frac{\left(-2-4\sqrt{5}\right)\left(5+2\sqrt{5}\right)}{25-4\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(-2-4\sqrt{5}\right)\left(5+2\sqrt{5}\right)}{25-4\times 5}$$

Multiply 4 and 5 to get 20.

$$\frac{\left(-2-4\sqrt{5}\right)\left(5+2\sqrt{5}\right)}{25-20}$$

Subtract 20 from 25 to get 5.

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

Apply the distributive property by multiplying each term of -2-4\sqrt{5} by each term of 5+2\sqrt{5}.

$$\frac{-10-4\sqrt{5}-20\sqrt{5}-8\left(\sqrt{5}\right)^{2}}{5}$$

Combine -4\sqrt{5} and -20\sqrt{5} to get -24\sqrt{5}.

$$\frac{-10-24\sqrt{5}-8\left(\sqrt{5}\right)^{2}}{5}$$

The square of \sqrt{5} is 5.

$$\frac{-10-24\sqrt{5}-8\times 5}{5}$$

Multiply -8 and 5 to get -40.

$$\frac{-10-24\sqrt{5}-40}{5}$$

Subtract 40 from -10 to get -50.

$$\frac{-50-24\sqrt{5}}{5}$$