Solve 20(n+2(n^2-n(n+1)/2))=3000

$$20 ( n + 2 ( n ^ { 2 } - \frac { n ( n + 1 ) } { 2 } ) ) = 3000$$

Solve for n

$$n=5\sqrt{6}\approx 12.247448714$$

$$n=-5\sqrt{6}\approx -12.247448714$$

Steps by Finding Square Root

Divide both sides by 20.

$$n+2\left(n^{2}-\frac{n\left(n+1\right)}{2}\right)=\frac{3000}{20}$$

Divide 3000 by 20 to get 150.

$$n+2\left(n^{2}-\frac{n\left(n+1\right)}{2}\right)=150$$

Multiply both sides of the equation by 2.

$$2n+4\left(n^{2}-\frac{n\left(n+1\right)}{2}\right)=300$$

Use the distributive property to multiply n by n+1.

$$2n+4\left(n^{2}-\frac{n^{2}+n}{2}\right)=300$$

Use the distributive property to multiply 4 by n^{2}-\frac{n^{2}+n}{2}.

$$2n+4n^{2}+4\left(-\frac{n^{2}+n}{2}\right)=300$$

Cancel out 2, the greatest common factor in 4 and 2.

$$2n+4n^{2}-2\left(n^{2}+n\right)=300$$

Use the distributive property to multiply -2 by n^{2}+n.

$$2n+4n^{2}-2n^{2}-2n=300$$

Combine 4n^{2} and -2n^{2} to get 2n^{2}.

$$2n+2n^{2}-2n=300$$

Combine 2n and -2n to get 0.

$$2n^{2}=300$$

Divide both sides by 2.

$$n^{2}=\frac{300}{2}$$

Divide 300 by 2 to get 150.

$$n^{2}=150$$

Take the square root of both sides of the equation.

$$n=5\sqrt{6} n=-5\sqrt{6}$$

Steps Using the Quadratic Formula

Divide both sides by 20.

$$n+2\left(n^{2}-\frac{n\left(n+1\right)}{2}\right)=\frac{3000}{20}$$

Divide 3000 by 20 to get 150.

$$n+2\left(n^{2}-\frac{n\left(n+1\right)}{2}\right)=150$$

Multiply both sides of the equation by 2.

$$2n+4\left(n^{2}-\frac{n\left(n+1\right)}{2}\right)=300$$

Use the distributive property to multiply n by n+1.

$$2n+4\left(n^{2}-\frac{n^{2}+n}{2}\right)=300$$

Use the distributive property to multiply 4 by n^{2}-\frac{n^{2}+n}{2}.

$$2n+4n^{2}+4\left(-\frac{n^{2}+n}{2}\right)=300$$

Cancel out 2, the greatest common factor in 4 and 2.

$$2n+4n^{2}-2\left(n^{2}+n\right)=300$$

Use the distributive property to multiply -2 by n^{2}+n.

$$2n+4n^{2}-2n^{2}-2n=300$$

Combine 4n^{2} and -2n^{2} to get 2n^{2}.

$$2n+2n^{2}-2n=300$$

Combine 2n and -2n to get 0.

$$2n^{2}=300$$

Subtract 300 from both sides.

$$2n^{2}-300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 0 for b, and -300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\times 2\left(-300\right)}}{2\times 2}$$

Square 0.

$$n=\frac{0±\sqrt{-4\times 2\left(-300\right)}}{2\times 2}$$

Multiply -4 times 2.

$$n=\frac{0±\sqrt{-8\left(-300\right)}}{2\times 2}$$

Multiply -8 times -300.

$$n=\frac{0±\sqrt{2400}}{2\times 2}$$

Take the square root of 2400.

$$n=\frac{0±20\sqrt{6}}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{0±20\sqrt{6}}{4}$$

Now solve the equation n=\frac{0±20\sqrt{6}}{4} when ± is plus.

$$n=5\sqrt{6}$$

Now solve the equation n=\frac{0±20\sqrt{6}}{4} when ± is minus.

$$n=-5\sqrt{6}$$

The equation is now solved.

$$n=5\sqrt{6} n=-5\sqrt{6}$$