Solve 30sqrt{3/9-2n}(3-n)=0

$$30 \sqrt { \frac { 3 } { 9 - 2 n } } ( 3 - n ) = 0$$

Solve for n

$$n=3$$

Solution Steps

Use the distributive property to multiply 30\sqrt{\frac{3}{9-2n}} by 3-n.

$$90\sqrt{\frac{3}{9-2n}}-30\sqrt{\frac{3}{9-2n}}n=0$$

Subtract -30\sqrt{\frac{3}{9-2n}}n from both sides of the equation.

$$90\sqrt{\frac{3}{9-2n}}=30\sqrt{\frac{3}{9-2n}}n$$

Square both sides of the equation.

$$\left(90\sqrt{\frac{3}{9-2n}}\right)^{2}=\left(30\sqrt{\frac{3}{9-2n}}n\right)^{2}$$

Expand \left(90\sqrt{\frac{3}{9-2n}}\right)^{2}.

$$90^{2}\left(\sqrt{\frac{3}{9-2n}}\right)^{2}=\left(30\sqrt{\frac{3}{9-2n}}n\right)^{2}$$

Calculate 90 to the power of 2 and get 8100.

$$8100\left(\sqrt{\frac{3}{9-2n}}\right)^{2}=\left(30\sqrt{\frac{3}{9-2n}}n\right)^{2}$$

Calculate \sqrt{\frac{3}{9-2n}} to the power of 2 and get \frac{3}{9-2n}.

$$8100\times \frac{3}{9-2n}=\left(30\sqrt{\frac{3}{9-2n}}n\right)^{2}$$

Express 8100\times \frac{3}{9-2n} as a single fraction.

$$\frac{8100\times 3}{9-2n}=\left(30\sqrt{\frac{3}{9-2n}}n\right)^{2}$$

Expand \left(30\sqrt{\frac{3}{9-2n}}n\right)^{2}.

$$\frac{8100\times 3}{9-2n}=30^{2}\left(\sqrt{\frac{3}{9-2n}}\right)^{2}n^{2}$$

Calculate 30 to the power of 2 and get 900.

$$\frac{8100\times 3}{9-2n}=900\left(\sqrt{\frac{3}{9-2n}}\right)^{2}n^{2}$$

Calculate \sqrt{\frac{3}{9-2n}} to the power of 2 and get \frac{3}{9-2n}.

$$\frac{8100\times 3}{9-2n}=900\times \frac{3}{9-2n}n^{2}$$

Express 900\times \frac{3}{9-2n} as a single fraction.

$$\frac{8100\times 3}{9-2n}=\frac{900\times 3}{9-2n}n^{2}$$

Express \frac{900\times 3}{9-2n}n^{2} as a single fraction.

$$\frac{8100\times 3}{9-2n}=\frac{900\times 3n^{2}}{9-2n}$$

Multiply 8100 and 3 to get 24300.

$$\frac{24300}{9-2n}=\frac{900\times 3n^{2}}{9-2n}$$

Multiply 900 and 3 to get 2700.

$$\frac{24300}{9-2n}=\frac{2700n^{2}}{9-2n}$$

Subtract \frac{2700n^{2}}{9-2n} from both sides.

$$\frac{24300}{9-2n}-\frac{2700n^{2}}{9-2n}=0$$

Since \frac{24300}{9-2n} and \frac{2700n^{2}}{9-2n} have the same denominator, subtract them by subtracting their numerators.

$$\frac{24300-2700n^{2}}{9-2n}=0$$

Variable n cannot be equal to \frac{9}{2} since division by zero is not defined. Multiply both sides of the equation by -2n+9.

$$24300-2700n^{2}=0$$

Subtract 24300 from both sides. Anything subtracted from zero gives its negation.

$$-2700n^{2}=-24300$$

Divide both sides by -2700.

$$n^{2}=\frac{-24300}{-2700}$$

Divide -24300 by -2700 to get 9.

$$n^{2}=9$$

Take the square root of both sides of the equation.

$$n=3 n=-3$$

Substitute 3 for n in the equation 30\sqrt{\frac{3}{9-2n}}\left(3-n\right)=0.

$$30\sqrt{\frac{3}{9-2\times 3}}\left(3-3\right)=0$$

Simplify. The value n=3 satisfies the equation.

$$0=0$$

Substitute -3 for n in the equation 30\sqrt{\frac{3}{9-2n}}\left(3-n\right)=0.

$$30\sqrt{\frac{3}{9-2\left(-3\right)}}\left(3-\left(-3\right)\right)=0$$

Simplify. The value n=-3 does not satisfy the equation.

$$36\times 5^{\frac{1}{2}}=0$$

Equation 90\sqrt{\frac{3}{9-2n}}=30\sqrt{\frac{3}{9-2n}}n has a unique solution.

$$n=3$$