Solve -5sqrt{2}-(sqrt{5}-sqrt{2})^2

$$- 5 \sqrt { 2 } - ( \sqrt { 5 } - \sqrt { 2 } ) ^ { 2 }$$

Evaluate

$$2\sqrt{10}-5\sqrt{2}-7\approx -7.746512492$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-\sqrt{2}\right)^{2}.

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

The square of \sqrt{5} is 5.

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

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

$$-5\sqrt{2}-\left(5-2\sqrt{10}+\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$-5\sqrt{2}-\left(5-2\sqrt{10}+2\right)$$

Add 5 and 2 to get 7.

$$-5\sqrt{2}-\left(7-2\sqrt{10}\right)$$

To find the opposite of 7-2\sqrt{10}, find the opposite of each term.

$$-5\sqrt{2}-7+2\sqrt{10}$$