Solve frac{-2sqrt{5}}{-2+sqrt{6}}

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

Evaluate

$$-\sqrt{5}\left(\sqrt{6}+2\right)\approx -9.94936153$$

Solution Steps

Rationalize the denominator of \frac{-2\sqrt{5}}{-2+\sqrt{6}} by multiplying numerator and denominator by -2-\sqrt{6}.

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

Consider \left(-2+\sqrt{6}\right)\left(-2-\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{-2\sqrt{5}\left(-2-\sqrt{6}\right)}{\left(-2\right)^{2}-\left(\sqrt{6}\right)^{2}}$$

Square -2. Square \sqrt{6}.

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

Subtract 6 from 4 to get -2.

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

Use the distributive property to multiply -2\sqrt{5} by -2-\sqrt{6}.

$$\frac{4\sqrt{5}+2\sqrt{5}\sqrt{6}}{-2}$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{4\sqrt{5}+2\sqrt{30}}{-2}$$

Divide each term of 4\sqrt{5}+2\sqrt{30} by -2 to get -2\sqrt{5}-\sqrt{30}.

$$-2\sqrt{5}-\sqrt{30}$$