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

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

Evaluate

$$\frac{\sqrt{5}-25}{20}\approx -1.138196601$$

Short Solution Steps

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

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

The square of \sqrt{5} is 5.

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

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

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

Consider \left(3\sqrt{5}-5\right)\left(3\sqrt{5}+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}.

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

Expand \left(3\sqrt{5}\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{5} is 5.

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

Multiply 9 and 5 to get 45.

$$\frac{3\sqrt{5}}{5}-\frac{\left(2+\sqrt{5}\right)\left(3\sqrt{5}+5\right)}{45-5^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{3\sqrt{5}}{5}-\frac{\left(2+\sqrt{5}\right)\left(3\sqrt{5}+5\right)}{45-25}$$

Subtract 25 from 45 to get 20.

$$\frac{3\sqrt{5}}{5}-\frac{\left(2+\sqrt{5}\right)\left(3\sqrt{5}+5\right)}{20}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 20 is 20. Multiply \frac{3\sqrt{5}}{5} times \frac{4}{4}.

$$\frac{4\times 3\sqrt{5}}{20}-\frac{\left(2+\sqrt{5}\right)\left(3\sqrt{5}+5\right)}{20}$$

Since \frac{4\times 3\sqrt{5}}{20} and \frac{\left(2+\sqrt{5}\right)\left(3\sqrt{5}+5\right)}{20} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\times 3\sqrt{5}-\left(2+\sqrt{5}\right)\left(3\sqrt{5}+5\right)}{20}$$

Do the multiplications in 4\times 3\sqrt{5}-\left(2+\sqrt{5}\right)\left(3\sqrt{5}+5\right).

$$\frac{12\sqrt{5}-6\sqrt{5}-10-15-5\sqrt{5}}{20}$$

Do the calculations in 12\sqrt{5}-6\sqrt{5}-10-15-5\sqrt{5}.

$$\frac{\sqrt{5}-25}{20}$$