Solve frac{3+5sqrt{5}}{3sqrt{5}}=a+bsqrt{5}+operatorname{hen}a+b=

$$\frac { 3 + 5 \sqrt { 5 } } { 3 \sqrt { 5 } } = a + b \sqrt { 5 } + \operatorname { hen } a + b =$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{15\sqrt{5}b+15b-3\sqrt{5}-25}{15\left(ehn+1\right)}\text{, }&n=0\text{ or }h\neq -\frac{1}{en}\\a\in \mathrm{C}\text{, }&b=\frac{11\sqrt{5}}{30}-\frac{1}{6}\text{ and }h=-\frac{1}{en}\text{ and }n\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{3+5\sqrt{5}}{3\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(3+5\sqrt{5}\right)\sqrt{5}}{3\left(\sqrt{5}\right)^{2}}=a+b\sqrt{5}+hena+b$$

The square of \sqrt{5} is 5.

$$\frac{\left(3+5\sqrt{5}\right)\sqrt{5}}{3\times 5}=a+b\sqrt{5}+hena+b$$

Multiply 3 and 5 to get 15.

$$\frac{\left(3+5\sqrt{5}\right)\sqrt{5}}{15}=a+b\sqrt{5}+hena+b$$

Use the distributive property to multiply 3+5\sqrt{5} by \sqrt{5}.

$$\frac{3\sqrt{5}+5\left(\sqrt{5}\right)^{2}}{15}=a+b\sqrt{5}+hena+b$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{5}+5\times 5}{15}=a+b\sqrt{5}+hena+b$$

Multiply 5 and 5 to get 25.

$$\frac{3\sqrt{5}+25}{15}=a+b\sqrt{5}+hena+b$$

Divide each term of 3\sqrt{5}+25 by 15 to get \frac{1}{5}\sqrt{5}+\frac{5}{3}.

$$\frac{1}{5}\sqrt{5}+\frac{5}{3}=a+b\sqrt{5}+hena+b$$

Swap sides so that all variable terms are on the left hand side.

$$a+b\sqrt{5}+hena+b=\frac{1}{5}\sqrt{5}+\frac{5}{3}$$

Subtract b\sqrt{5} from both sides.

$$a+hena+b=\frac{1}{5}\sqrt{5}+\frac{5}{3}-b\sqrt{5}$$

Subtract b from both sides.

$$a+hena=\frac{1}{5}\sqrt{5}+\frac{5}{3}-b\sqrt{5}-b$$

Reorder the terms.

$$eahn+a=-\sqrt{5}b-b+\frac{1}{5}\sqrt{5}+\frac{5}{3}$$

Combine all terms containing a.

$$\left(ehn+1\right)a=-\sqrt{5}b-b+\frac{1}{5}\sqrt{5}+\frac{5}{3}$$

The equation is in standard form.

$$\left(ehn+1\right)a=-\sqrt{5}b-b+\frac{\sqrt{5}}{5}+\frac{5}{3}$$

Divide both sides by 1+neh.

$$\frac{\left(ehn+1\right)a}{ehn+1}=\frac{-\sqrt{5}b-b+\frac{\sqrt{5}}{5}+\frac{5}{3}}{ehn+1}$$

Dividing by 1+neh undoes the multiplication by 1+neh.

$$a=\frac{-\sqrt{5}b-b+\frac{\sqrt{5}}{5}+\frac{5}{3}}{ehn+1}$$

Divide -\sqrt{5}b-b+\frac{\sqrt{5}}{5}+\frac{5}{3} by 1+neh.

$$a=\frac{-15\sqrt{5}b-15b+3\sqrt{5}+25}{15\left(ehn+1\right)}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{15\sqrt{5}b+15b-3\sqrt{5}-25}{15\left(ehn+1\right)}\text{, }&n=0\text{ or }h\neq -\frac{1}{en}\\a\in \mathrm{R}\text{, }&b=\frac{11\sqrt{5}}{30}-\frac{1}{6}\text{ and }h=-\frac{1}{en}\text{ and }n\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{3+5\sqrt{5}}{3\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(3+5\sqrt{5}\right)\sqrt{5}}{3\left(\sqrt{5}\right)^{2}}=a+b\sqrt{5}+hena+b$$

The square of \sqrt{5} is 5.

$$\frac{\left(3+5\sqrt{5}\right)\sqrt{5}}{3\times 5}=a+b\sqrt{5}+hena+b$$

Multiply 3 and 5 to get 15.

$$\frac{\left(3+5\sqrt{5}\right)\sqrt{5}}{15}=a+b\sqrt{5}+hena+b$$

Use the distributive property to multiply 3+5\sqrt{5} by \sqrt{5}.

$$\frac{3\sqrt{5}+5\left(\sqrt{5}\right)^{2}}{15}=a+b\sqrt{5}+hena+b$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{5}+5\times 5}{15}=a+b\sqrt{5}+hena+b$$

Multiply 5 and 5 to get 25.

$$\frac{3\sqrt{5}+25}{15}=a+b\sqrt{5}+hena+b$$

Divide each term of 3\sqrt{5}+25 by 15 to get \frac{1}{5}\sqrt{5}+\frac{5}{3}.

$$\frac{1}{5}\sqrt{5}+\frac{5}{3}=a+b\sqrt{5}+hena+b$$

Swap sides so that all variable terms are on the left hand side.

$$a+b\sqrt{5}+hena+b=\frac{1}{5}\sqrt{5}+\frac{5}{3}$$

Subtract b\sqrt{5} from both sides.

$$a+hena+b=\frac{1}{5}\sqrt{5}+\frac{5}{3}-b\sqrt{5}$$

Subtract b from both sides.

$$a+hena=\frac{1}{5}\sqrt{5}+\frac{5}{3}-b\sqrt{5}-b$$

Reorder the terms.

$$eahn+a=-\sqrt{5}b-b+\frac{1}{5}\sqrt{5}+\frac{5}{3}$$

Combine all terms containing a.

$$\left(ehn+1\right)a=-\sqrt{5}b-b+\frac{1}{5}\sqrt{5}+\frac{5}{3}$$

The equation is in standard form.

$$\left(ehn+1\right)a=-\sqrt{5}b-b+\frac{\sqrt{5}}{5}+\frac{5}{3}$$

Divide both sides by 1+hen.

$$\frac{\left(ehn+1\right)a}{ehn+1}=\frac{-\sqrt{5}b-b+\frac{\sqrt{5}}{5}+\frac{5}{3}}{ehn+1}$$

Dividing by 1+hen undoes the multiplication by 1+hen.

$$a=\frac{-\sqrt{5}b-b+\frac{\sqrt{5}}{5}+\frac{5}{3}}{ehn+1}$$

Divide -\sqrt{5}b-b+\frac{\sqrt{5}}{5}+\frac{5}{3} by 1+hen.

$$a=\frac{-15\sqrt{5}b-15b+3\sqrt{5}+25}{15\left(ehn+1\right)}$$

Solve for b

$$b=-\frac{\left(\sqrt{5}-1\right)\left(15eahn+15a-3\sqrt{5}-25\right)}{60}$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{3+5\sqrt{5}}{3\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(3+5\sqrt{5}\right)\sqrt{5}}{3\left(\sqrt{5}\right)^{2}}=a+b\sqrt{5}+hena+b$$

The square of \sqrt{5} is 5.

$$\frac{\left(3+5\sqrt{5}\right)\sqrt{5}}{3\times 5}=a+b\sqrt{5}+hena+b$$

Multiply 3 and 5 to get 15.

$$\frac{\left(3+5\sqrt{5}\right)\sqrt{5}}{15}=a+b\sqrt{5}+hena+b$$

Use the distributive property to multiply 3+5\sqrt{5} by \sqrt{5}.

$$\frac{3\sqrt{5}+5\left(\sqrt{5}\right)^{2}}{15}=a+b\sqrt{5}+hena+b$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{5}+5\times 5}{15}=a+b\sqrt{5}+hena+b$$

Multiply 5 and 5 to get 25.

$$\frac{3\sqrt{5}+25}{15}=a+b\sqrt{5}+hena+b$$

Divide each term of 3\sqrt{5}+25 by 15 to get \frac{1}{5}\sqrt{5}+\frac{5}{3}.

$$\frac{1}{5}\sqrt{5}+\frac{5}{3}=a+b\sqrt{5}+hena+b$$

Swap sides so that all variable terms are on the left hand side.

$$a+b\sqrt{5}+hena+b=\frac{1}{5}\sqrt{5}+\frac{5}{3}$$

Subtract a from both sides.

$$b\sqrt{5}+hena+b=\frac{1}{5}\sqrt{5}+\frac{5}{3}-a$$

Subtract hena from both sides.

$$b\sqrt{5}+b=\frac{1}{5}\sqrt{5}+\frac{5}{3}-a-hena$$

Reorder the terms.

$$\sqrt{5}b+b=-eahn-a+\frac{1}{5}\sqrt{5}+\frac{5}{3}$$

Combine all terms containing b.

$$\left(\sqrt{5}+1\right)b=-eahn-a+\frac{1}{5}\sqrt{5}+\frac{5}{3}$$

The equation is in standard form.

$$\left(\sqrt{5}+1\right)b=-eahn-a+\frac{\sqrt{5}}{5}+\frac{5}{3}$$

Divide both sides by \sqrt{5}+1.

$$\frac{\left(\sqrt{5}+1\right)b}{\sqrt{5}+1}=\frac{-eahn-a+\frac{\sqrt{5}}{5}+\frac{5}{3}}{\sqrt{5}+1}$$

Dividing by \sqrt{5}+1 undoes the multiplication by \sqrt{5}+1.

$$b=\frac{-eahn-a+\frac{\sqrt{5}}{5}+\frac{5}{3}}{\sqrt{5}+1}$$

Divide -a+\frac{\sqrt{5}}{5}-enha+\frac{5}{3} by \sqrt{5}+1.

$$b=\frac{\left(\sqrt{5}-1\right)\left(-15eahn-15a+3\sqrt{5}+25\right)}{60}$$