Solve 2/2-sqrt{5}-2/2+sqrt{5}

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

Evaluate

$$-4\sqrt{5}\approx -8.94427191$$

Solution Steps

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

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

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

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

Square 2. Square \sqrt{5}.

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

Subtract 5 from 4 to get -1.

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

Anything divided by -1 gives its opposite.

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

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

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

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

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

Square 2. Square \sqrt{5}.

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

Subtract 5 from 4 to get -1.

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

Anything divided by -1 gives its opposite.

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

The opposite of -2\left(2-\sqrt{5}\right) is 2\left(2-\sqrt{5}\right).

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

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

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

Use the distributive property to multiply 2 by 2-\sqrt{5}.

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

Add -4 and 4 to get 0.

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

Combine -2\sqrt{5} and -2\sqrt{5} to get -4\sqrt{5}.

$$-4\sqrt{5}$$