Solve n=n/3+15+2sqrt{n}

$$n= \frac{ n }{ 3 } +15+2 \sqrt{ n }$$

Solve for n

$$n = \frac{9 {(\sqrt{11} + 6)}}{2} \approx 41.924811557$$

Solution Steps

Subtract \frac{n}{3}+15 from both sides of the equation.

$$n-\left(\frac{n}{3}+15\right)=2\sqrt{n}$$

Multiply both sides of the equation by 3.

$$3n-3\left(\frac{n}{3}+15\right)=6\sqrt{n}$$

Use the distributive property to multiply -3 by \frac{n}{3}+15.

$$3n-3\times \frac{n}{3}-45=6\sqrt{n}$$

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

$$3n+\frac{-3n}{3}-45=6\sqrt{n}$$

Cancel out 3 and 3.

$$3n-n-45=6\sqrt{n}$$

Combine 3n and -n to get 2n.

$$2n-45=6\sqrt{n}$$

Square both sides of the equation.

$$\left(2n-45\right)^{2}=\left(6\sqrt{n}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2n-45\right)^{2}.

$$4n^{2}-180n+2025=\left(6\sqrt{n}\right)^{2}$$

Expand \left(6\sqrt{n}\right)^{2}.

$$4n^{2}-180n+2025=6^{2}\left(\sqrt{n}\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$4n^{2}-180n+2025=36\left(\sqrt{n}\right)^{2}$$

Calculate \sqrt{n} to the power of 2 and get n.

$$4n^{2}-180n+2025=36n$$

Subtract 36n from both sides.

$$4n^{2}-180n+2025-36n=0$$

Combine -180n and -36n to get -216n.

$$4n^{2}-216n+2025=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -216 for b, and 2025 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-216\right)±\sqrt{\left(-216\right)^{2}-4\times 4\times 2025}}{2\times 4}$$

Square -216.

$$n=\frac{-\left(-216\right)±\sqrt{46656-4\times 4\times 2025}}{2\times 4}$$

Multiply -4 times 4.

$$n=\frac{-\left(-216\right)±\sqrt{46656-16\times 2025}}{2\times 4}$$

Multiply -16 times 2025.

$$n=\frac{-\left(-216\right)±\sqrt{46656-32400}}{2\times 4}$$

Add 46656 to -32400.

$$n=\frac{-\left(-216\right)±\sqrt{14256}}{2\times 4}$$

Take the square root of 14256.

$$n=\frac{-\left(-216\right)±36\sqrt{11}}{2\times 4}$$

The opposite of -216 is 216.

$$n=\frac{216±36\sqrt{11}}{2\times 4}$$

Multiply 2 times 4.

$$n=\frac{216±36\sqrt{11}}{8}$$

Now solve the equation n=\frac{216±36\sqrt{11}}{8} when ± is plus. Add 216 to 36\sqrt{11}.

$$n=\frac{36\sqrt{11}+216}{8}$$

Divide 216+36\sqrt{11} by 8.

$$n=\frac{9\sqrt{11}}{2}+27$$

Now solve the equation n=\frac{216±36\sqrt{11}}{8} when ± is minus. Subtract 36\sqrt{11} from 216.

$$n=\frac{216-36\sqrt{11}}{8}$$

Divide 216-36\sqrt{11} by 8.

$$n=-\frac{9\sqrt{11}}{2}+27$$

The equation is now solved.

$$n=\frac{9\sqrt{11}}{2}+27 n=-\frac{9\sqrt{11}}{2}+27$$

Substitute \frac{9\sqrt{11}}{2}+27 for n in the equation n=\frac{n}{3}+15+2\sqrt{n}.

$$\frac{9\sqrt{11}}{2}+27=\frac{\frac{9\sqrt{11}}{2}+27}{3}+15+2\sqrt{\frac{9\sqrt{11}}{2}+27}$$

Simplify. The value n=\frac{9\sqrt{11}}{2}+27 satisfies the equation.

$$\frac{9}{2}\times 11^{\frac{1}{2}}+27=\frac{9}{2}\times 11^{\frac{1}{2}}+27$$

Substitute -\frac{9\sqrt{11}}{2}+27 for n in the equation n=\frac{n}{3}+15+2\sqrt{n}.

$$-\frac{9\sqrt{11}}{2}+27=\frac{-\frac{9\sqrt{11}}{2}+27}{3}+15+2\sqrt{-\frac{9\sqrt{11}}{2}+27}$$

Simplify. The value n=-\frac{9\sqrt{11}}{2}+27 does not satisfy the equation.

$$-\frac{9}{2}\times 11^{\frac{1}{2}}+27=\frac{3}{2}\times 11^{\frac{1}{2}}+21$$

Equation 2n-45=6\sqrt{n} has a unique solution.

$$n=\frac{9\sqrt{11}}{2}+27$$