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

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

Evaluate

$$\frac{-31\sqrt{5}-63}{11}\approx -12.028918846$$

Short Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{5}.

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

Subtract 5 from 4 to get -1.

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

Anything divided by -1 gives its opposite.

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

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

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

Consider \left(4+\sqrt{5}\right)\left(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}.

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

Square 4. Square \sqrt{5}.

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

Subtract 5 from 16 to get 11.

$$-3\left(2+\sqrt{5}\right)+\frac{\left(2+\sqrt{5}\right)\left(4-\sqrt{5}\right)}{11}$$

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

$$-6-3\sqrt{5}+\frac{\left(2+\sqrt{5}\right)\left(4-\sqrt{5}\right)}{11}$$

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

$$-6-3\sqrt{5}+\frac{8-2\sqrt{5}+4\sqrt{5}-\left(\sqrt{5}\right)^{2}}{11}$$

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

$$-6-3\sqrt{5}+\frac{8+2\sqrt{5}-\left(\sqrt{5}\right)^{2}}{11}$$

The square of \sqrt{5} is 5.

$$-6-3\sqrt{5}+\frac{8+2\sqrt{5}-5}{11}$$

Subtract 5 from 8 to get 3.

$$-6-3\sqrt{5}+\frac{3+2\sqrt{5}}{11}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -6-3\sqrt{5} times \frac{11}{11}.

$$\frac{11\left(-6-3\sqrt{5}\right)}{11}+\frac{3+2\sqrt{5}}{11}$$

Since \frac{11\left(-6-3\sqrt{5}\right)}{11} and \frac{3+2\sqrt{5}}{11} have the same denominator, add them by adding their numerators.

$$\frac{11\left(-6-3\sqrt{5}\right)+3+2\sqrt{5}}{11}$$

Do the multiplications in 11\left(-6-3\sqrt{5}\right)+3+2\sqrt{5}.

$$\frac{-66-33\sqrt{5}+3+2\sqrt{5}}{11}$$

Do the calculations in -66-33\sqrt{5}+3+2\sqrt{5}.

$$\frac{-63-31\sqrt{5}}{11}$$