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

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

Solve for P

$$P=-13\sqrt{3}-7\approx -29.516660498$$

Steps for Solving Linear Equation

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

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

Add 9 and 12 to get 21.

$$\frac{21+12\sqrt{3}}{3^{2}-\left(-2\sqrt{3}\right)^{2}}=P+9\sqrt{3}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{21+12\sqrt{3}}{9-\left(-2\sqrt{3}\right)^{2}}=P+9\sqrt{3}$$

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

$$\frac{21+12\sqrt{3}}{9-\left(-2\right)^{2}\left(\sqrt{3}\right)^{2}}=P+9\sqrt{3}$$

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

$$\frac{21+12\sqrt{3}}{9-4\left(\sqrt{3}\right)^{2}}=P+9\sqrt{3}$$

The square of \sqrt{3} is 3.

$$\frac{21+12\sqrt{3}}{9-4\times 3}=P+9\sqrt{3}$$

Multiply 4 and 3 to get 12.

$$\frac{21+12\sqrt{3}}{9-12}=P+9\sqrt{3}$$

Subtract 12 from 9 to get -3.

$$\frac{21+12\sqrt{3}}{-3}=P+9\sqrt{3}$$

Divide each term of 21+12\sqrt{3} by -3 to get -7-4\sqrt{3}.

$$-7-4\sqrt{3}=P+9\sqrt{3}$$

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

$$P+9\sqrt{3}=-7-4\sqrt{3}$$

Subtract 9\sqrt{3} from both sides.

$$P=-7-4\sqrt{3}-9\sqrt{3}$$

Combine -4\sqrt{3} and -9\sqrt{3} to get -13\sqrt{3}.

$$P=-7-13\sqrt{3}$$