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200\times 2=n\left(3n+1\right)
Multiply both sides by 2.
400=n\left(3n+1\right)
Multiply 200 and 2 to get 400.
400=3n^{2}+n
Use the distributive property to multiply n by 3n+1.
3n^{2}+n=400
Swap sides so that all variable terms are on the left hand side.
3n^{2}+n-400=0
Subtract 400 from both sides.
n=\frac{-1±\sqrt{1^{2}-4\times 3\left(-400\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 1 for b, and -400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
n=\frac{-1±\sqrt{1-4\times 3\left(-400\right)}}{2\times 3}
Square 1.
n=\frac{-1±\sqrt{1-12\left(-400\right)}}{2\times 3}
Multiply -4 times 3.
n=\frac{-1±\sqrt{1+4800}}{2\times 3}
Multiply -12 times -400.
n=\frac{-1±\sqrt{4801}}{2\times 3}
Add 1 to 4800.
n=\frac{-1±\sqrt{4801}}{6}
Multiply 2 times 3.
n=\frac{\sqrt{4801}-1}{6}
Now solve the equation n=\frac{-1±\sqrt{4801}}{6} when ± is plus. Add -1 to \sqrt{4801}.
n=\frac{-\sqrt{4801}-1}{6}
Now solve the equation n=\frac{-1±\sqrt{4801}}{6} when ± is minus. Subtract \sqrt{4801} from -1.
n=\frac{\sqrt{4801}-1}{6} n=\frac{-\sqrt{4801}-1}{6}
The equation is now solved.
200\times 2=n\left(3n+1\right)
Multiply both sides by 2.
400=n\left(3n+1\right)
Multiply 200 and 2 to get 400.
400=3n^{2}+n
Use the distributive property to multiply n by 3n+1.
3n^{2}+n=400
Swap sides so that all variable terms are on the left hand side.
\frac{3n^{2}+n}{3}=\frac{400}{3}
Divide both sides by 3.
n^{2}+\frac{1}{3}n=\frac{400}{3}
Dividing by 3 undoes the multiplication by 3.
n^{2}+\frac{1}{3}n+\left(\frac{1}{6}\right)^{2}=\frac{400}{3}+\left(\frac{1}{6}\right)^{2}
Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
n^{2}+\frac{1}{3}n+\frac{1}{36}=\frac{400}{3}+\frac{1}{36}
Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.
n^{2}+\frac{1}{3}n+\frac{1}{36}=\frac{4801}{36}
Add \frac{400}{3} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(n+\frac{1}{6}\right)^{2}=\frac{4801}{36}
Factor n^{2}+\frac{1}{3}n+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(n+\frac{1}{6}\right)^{2}}=\sqrt{\frac{4801}{36}}
Take the square root of both sides of the equation.
n+\frac{1}{6}=\frac{\sqrt{4801}}{6} n+\frac{1}{6}=-\frac{\sqrt{4801}}{6}
Simplify.
n=\frac{\sqrt{4801}-1}{6} n=\frac{-\sqrt{4801}-1}{6}
Subtract \frac{1}{6} from both sides of the equation.