Solve 2*2/-1+sqrt{5}

$$2 \times \frac{ 2 }{ -1+ \sqrt{ 5 } }$$

Evaluate

$$\sqrt{5}+1\approx 3.236067977$$

Solution Steps

Rationalize the denominator of \frac{2}{-1+\sqrt{5}} by multiplying numerator and denominator by -1-\sqrt{5}.

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

Consider \left(-1+\sqrt{5}\right)\left(-1-\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}.

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

Square -1. Square \sqrt{5}.

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

Subtract 5 from 1 to get -4.

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

Divide 2\left(-1-\sqrt{5}\right) by -4 to get -\frac{1}{2}\left(-1-\sqrt{5}\right).

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

Use the distributive property to multiply -\frac{1}{2} by -1-\sqrt{5}.

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

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

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

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

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

Use the distributive property to multiply 2 by \frac{1}{2}+\frac{1}{2}\sqrt{5}.

$$2\times \frac{1}{2}+2\times \frac{1}{2}\sqrt{5}$$

Cancel out 2 and 2.

$$1+2\times \frac{1}{2}\sqrt{5}$$

Cancel out 2 and 2.

$$1+\sqrt{5}$$

Factor

$$\sqrt{5} + 1 = 3.236067977$$