Solve frac{sqrt{5}+sqrt{3}}{sqrt{5}-sqrt{3}}=a+bsqrt{15}

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

Solve for b

$$b=-\frac{\sqrt{15}\left(a-\sqrt{15}-4\right)}{15}$$

Steps for Solving Linear Equation

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

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

Consider \left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{5}. Square \sqrt{3}.

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

Subtract 3 from 5 to get 2.

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

Multiply \sqrt{5}+\sqrt{3} and \sqrt{5}+\sqrt{3} to get \left(\sqrt{5}+\sqrt{3}\right)^{2}.

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

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

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

The square of \sqrt{5} is 5.

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

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

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

The square of \sqrt{3} is 3.

$$\frac{5+2\sqrt{15}+3}{2}=a+b\sqrt{15}$$

Add 5 and 3 to get 8.

$$\frac{8+2\sqrt{15}}{2}=a+b\sqrt{15}$$

Divide each term of 8+2\sqrt{15} by 2 to get 4+\sqrt{15}.

$$4+\sqrt{15}=a+b\sqrt{15}$$

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

$$a+b\sqrt{15}=4+\sqrt{15}$$

Subtract a from both sides.

$$b\sqrt{15}=4+\sqrt{15}-a$$

The equation is in standard form.

$$\sqrt{15}b=-a+\sqrt{15}+4$$

Divide both sides by \sqrt{15}.

$$\frac{\sqrt{15}b}{\sqrt{15}}=\frac{-a+\sqrt{15}+4}{\sqrt{15}}$$

Dividing by \sqrt{15} undoes the multiplication by \sqrt{15}.

$$b=\frac{-a+\sqrt{15}+4}{\sqrt{15}}$$

Divide 4+\sqrt{15}-a by \sqrt{15}.

$$b=\frac{\sqrt{15}\left(-a+\sqrt{15}+4\right)}{15}$$

Solve for a

$$a=-\sqrt{15}b+\sqrt{15}+4$$