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

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

Evaluate

$$22-4\sqrt{10}\approx 9.350889359$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

Multiply 4 and 5 to get 20.

$$20-4\sqrt{5}\sqrt{2}+\left(\sqrt{2}\right)^{2}$$

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

$$20-4\sqrt{10}+\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$20-4\sqrt{10}+2$$

Add 20 and 2 to get 22.

$$22-4\sqrt{10}$$

Expand

$$22-4\sqrt{10}$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

Multiply 4 and 5 to get 20.

$$20-4\sqrt{5}\sqrt{2}+\left(\sqrt{2}\right)^{2}$$

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

$$20-4\sqrt{10}+\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$20-4\sqrt{10}+2$$

Add 20 and 2 to get 22.

$$22-4\sqrt{10}$$