Solve {left(sqrt{5}-2right)}^2+{left(sqrt{5}-2right)}^2-2(sqrt{5}-2)(sqrt{5}+3)

$${ \left( \sqrt{ 5 } -2 \right) }^{ 2 } + { \left( \sqrt{ 5 } -2 \right) }^{ 2 } -2( \sqrt{ 5 } -2)( \sqrt{ 5 } +3)$$

Evaluate

$$20-10\sqrt{5}\approx -2.360679775$$

Solution Steps

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

$$\left(\sqrt{5}\right)^{2}-4\sqrt{5}+4+\left(\sqrt{5}-2\right)^{2}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

The square of \sqrt{5} is 5.

$$5-4\sqrt{5}+4+\left(\sqrt{5}-2\right)^{2}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Add 5 and 4 to get 9.

$$9-4\sqrt{5}+\left(\sqrt{5}-2\right)^{2}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

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

$$9-4\sqrt{5}+\left(\sqrt{5}\right)^{2}-4\sqrt{5}+4-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

The square of \sqrt{5} is 5.

$$9-4\sqrt{5}+5-4\sqrt{5}+4-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Add 5 and 4 to get 9.

$$9-4\sqrt{5}+9-4\sqrt{5}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Add 9 and 9 to get 18.

$$18-4\sqrt{5}-4\sqrt{5}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Combine -4\sqrt{5} and -4\sqrt{5} to get -8\sqrt{5}.

$$18-8\sqrt{5}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

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

$$18-8\sqrt{5}+\left(-2\sqrt{5}+4\right)\left(\sqrt{5}+3\right)$$

Use the distributive property to multiply -2\sqrt{5}+4 by \sqrt{5}+3 and combine like terms.

$$18-8\sqrt{5}-2\left(\sqrt{5}\right)^{2}-2\sqrt{5}+12$$

The square of \sqrt{5} is 5.

$$18-8\sqrt{5}-2\times 5-2\sqrt{5}+12$$

Multiply -2 and 5 to get -10.

$$18-8\sqrt{5}-10-2\sqrt{5}+12$$

Add -10 and 12 to get 2.

$$18-8\sqrt{5}+2-2\sqrt{5}$$

Add 18 and 2 to get 20.

$$20-8\sqrt{5}-2\sqrt{5}$$

Combine -8\sqrt{5} and -2\sqrt{5} to get -10\sqrt{5}.

$$20-10\sqrt{5}$$

Factor

$$10 {(2 - \sqrt{5})} = -2.360679775$$