Solve {c}{(2)quad(sqrt{20}-sqrt{45})divsqrt{5}(3)quad(sqrt{5}-2)^2+(sqrt{5}}{+3)^2-2(sqrt{5}-2)quad(sqrt{5}+3)}

$$\left. \begin{array} { c } { ( 2 ) \quad ( \sqrt { 20 } - \sqrt { 45 } ) \div \sqrt { 5 } ( 3 ) \quad ( \sqrt { 5 } - 2 ) ^ { 2 } + ( \sqrt { 5 } } \\ { + 3 ) ^ { 2 } - 2 ( \sqrt { 5 } - 2 ) \quad ( \sqrt { 5 } + 3 ) } \end{array} \right.$$

Evaluate

$$28\sqrt{5}-38\approx 24.60990337$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{2\left(2\sqrt{5}-\sqrt{45}\right)}{\sqrt{5}}\times 3\left(\sqrt{5}-2\right)^{2}+\left(\sqrt{5}+3\right)^{2}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

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

Combine 2\sqrt{5} and -3\sqrt{5} to get -\sqrt{5}.

$$\frac{2\left(-1\right)\sqrt{5}}{\sqrt{5}}\times 3\left(\sqrt{5}-2\right)^{2}+\left(\sqrt{5}+3\right)^{2}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Multiply 2 and -1 to get -2.

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

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

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

The square of \sqrt{5} is 5.

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

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

Multiply -2 and 5 to get -10.

$$\frac{-10}{5}\times 3\left(\sqrt{5}-2\right)^{2}+\left(\sqrt{5}+3\right)^{2}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Divide -10 by 5 to get -2.

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

Multiply -2 and 3 to get -6.

$$-6\left(\sqrt{5}-2\right)^{2}+\left(\sqrt{5}+3\right)^{2}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

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

$$-6\left(\left(\sqrt{5}\right)^{2}-4\sqrt{5}+4\right)+\left(\sqrt{5}+3\right)^{2}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

The square of \sqrt{5} is 5.

$$-6\left(5-4\sqrt{5}+4\right)+\left(\sqrt{5}+3\right)^{2}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Add 5 and 4 to get 9.

$$-6\left(9-4\sqrt{5}\right)+\left(\sqrt{5}+3\right)^{2}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Use the distributive property to multiply -6 by 9-4\sqrt{5}.

$$-54+24\sqrt{5}+\left(\sqrt{5}+3\right)^{2}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{5}+3\right)^{2}.

$$-54+24\sqrt{5}+\left(\sqrt{5}\right)^{2}+6\sqrt{5}+9-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

The square of \sqrt{5} is 5.

$$-54+24\sqrt{5}+5+6\sqrt{5}+9-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Add 5 and 9 to get 14.

$$-54+24\sqrt{5}+14+6\sqrt{5}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Add -54 and 14 to get -40.

$$-40+24\sqrt{5}+6\sqrt{5}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

Combine 24\sqrt{5} and 6\sqrt{5} to get 30\sqrt{5}.

$$-40+30\sqrt{5}-2\left(\sqrt{5}-2\right)\left(\sqrt{5}+3\right)$$

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

$$-40+30\sqrt{5}+\left(-2\sqrt{5}+4\right)\left(\sqrt{5}+3\right)$$

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

$$-40+30\sqrt{5}-2\left(\sqrt{5}\right)^{2}-2\sqrt{5}+12$$

The square of \sqrt{5} is 5.

$$-40+30\sqrt{5}-2\times 5-2\sqrt{5}+12$$

Multiply -2 and 5 to get -10.

$$-40+30\sqrt{5}-10-2\sqrt{5}+12$$

Add -10 and 12 to get 2.

$$-40+30\sqrt{5}+2-2\sqrt{5}$$

Add -40 and 2 to get -38.

$$-38+30\sqrt{5}-2\sqrt{5}$$

Combine 30\sqrt{5} and -2\sqrt{5} to get 28\sqrt{5}.

$$-38+28\sqrt{5}$$