Solve (1/2+3sqrt{5})(2sqrt{2}-4)

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

Evaluate

$$\sqrt{2}+6\sqrt{10}-12\sqrt{5}-2\approx -8.444936207$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{1}{2}+3\sqrt{5} by each term of 2\sqrt{2}-4.

$$\frac{1}{2}\times 2\sqrt{2}+\frac{1}{2}\left(-4\right)+6\sqrt{5}\sqrt{2}-12\sqrt{5}$$

Cancel out 2 and 2.

$$\sqrt{2}+\frac{1}{2}\left(-4\right)+6\sqrt{5}\sqrt{2}-12\sqrt{5}$$

Multiply \frac{1}{2} and -4 to get \frac{-4}{2}.

$$\sqrt{2}+\frac{-4}{2}+6\sqrt{5}\sqrt{2}-12\sqrt{5}$$

Divide -4 by 2 to get -2.

$$\sqrt{2}-2+6\sqrt{5}\sqrt{2}-12\sqrt{5}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\sqrt{2}-2+6\sqrt{10}-12\sqrt{5}$$

Factor

$$\sqrt{2} + 6 \sqrt{10} - 12 \sqrt{5} - 2 = -8.444936207$$