Solve -1/2sqrt{5+1}

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

Evaluate

$$\frac{1-2\sqrt{5}}{19}\approx -0.182743998$$

Solution Steps

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

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

Consider \left(2\sqrt{5}+1\right)\left(2\sqrt{5}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{5} is 5.

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

Multiply 4 and 5 to get 20.

$$\frac{-\left(2\sqrt{5}-1\right)}{20-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{-\left(2\sqrt{5}-1\right)}{20-1}$$

Subtract 1 from 20 to get 19.

$$\frac{-\left(2\sqrt{5}-1\right)}{19}$$

To find the opposite of 2\sqrt{5}-1, find the opposite of each term.

$$\frac{-2\sqrt{5}-\left(-1\right)}{19}$$

The opposite of -1 is 1.

$$\frac{-2\sqrt{5}+1}{19}$$