Solve frac{3+2sqrt{3}}{2-2sqrt{3}}=P+9sqrt{3}

$$\frac { 3 + 2 \sqrt { 3 } } { 2 - 2 \sqrt { 3 } } = P + 9 \sqrt { 3 }$$

Solve for P

$$P=\frac{-41\sqrt{3}-9}{4}\approx -20.003520778$$

Steps for Solving Linear Equation

Rationalize the denominator of \frac{3+2\sqrt{3}}{2-2\sqrt{3}} by multiplying numerator and denominator by 2+2\sqrt{3}.

$$\frac{\left(3+2\sqrt{3}\right)\left(2+2\sqrt{3}\right)}{\left(2-2\sqrt{3}\right)\left(2+2\sqrt{3}\right)}=P+9\sqrt{3}$$

Consider \left(2-2\sqrt{3}\right)\left(2+2\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}.

$$\frac{\left(3+2\sqrt{3}\right)\left(2+2\sqrt{3}\right)}{2^{2}-\left(-2\sqrt{3}\right)^{2}}=P+9\sqrt{3}$$

Calculate 2 to the power of 2 and get 4.

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

Expand \left(-2\sqrt{3}\right)^{2}.

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

Calculate -2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

$$\frac{\left(3+2\sqrt{3}\right)\left(2+2\sqrt{3}\right)}{4-4\times 3}=P+9\sqrt{3}$$

Multiply 4 and 3 to get 12.

$$\frac{\left(3+2\sqrt{3}\right)\left(2+2\sqrt{3}\right)}{4-12}=P+9\sqrt{3}$$

Subtract 12 from 4 to get -8.

$$\frac{\left(3+2\sqrt{3}\right)\left(2+2\sqrt{3}\right)}{-8}=P+9\sqrt{3}$$

Apply the distributive property by multiplying each term of 3+2\sqrt{3} by each term of 2+2\sqrt{3}.

$$\frac{6+6\sqrt{3}+4\sqrt{3}+4\left(\sqrt{3}\right)^{2}}{-8}=P+9\sqrt{3}$$

Combine 6\sqrt{3} and 4\sqrt{3} to get 10\sqrt{3}.

$$\frac{6+10\sqrt{3}+4\left(\sqrt{3}\right)^{2}}{-8}=P+9\sqrt{3}$$

The square of \sqrt{3} is 3.

$$\frac{6+10\sqrt{3}+4\times 3}{-8}=P+9\sqrt{3}$$

Multiply 4 and 3 to get 12.

$$\frac{6+10\sqrt{3}+12}{-8}=P+9\sqrt{3}$$

Add 6 and 12 to get 18.

$$\frac{18+10\sqrt{3}}{-8}=P+9\sqrt{3}$$

Divide each term of 18+10\sqrt{3} by -8 to get -\frac{9}{4}-\frac{5}{4}\sqrt{3}.

$$-\frac{9}{4}-\frac{5}{4}\sqrt{3}=P+9\sqrt{3}$$

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

$$P+9\sqrt{3}=-\frac{9}{4}-\frac{5}{4}\sqrt{3}$$

Subtract 9\sqrt{3} from both sides.

$$P=-\frac{9}{4}-\frac{5}{4}\sqrt{3}-9\sqrt{3}$$

Combine -\frac{5}{4}\sqrt{3} and -9\sqrt{3} to get -\frac{41}{4}\sqrt{3}.

$$P=-\frac{9}{4}-\frac{41}{4}\sqrt{3}$$