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n\left(n+1\right)=50\times 2
Multiply both sides by 2.
n^{2}+n=50\times 2
Use the distributive property to multiply n by n+1.
n^{2}+n=100
Multiply 50 and 2 to get 100.
n^{2}+n-100=0
Subtract 100 from both sides.
n=\frac{-1±\sqrt{1^{2}-4\left(-100\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
n=\frac{-1±\sqrt{1-4\left(-100\right)}}{2}
Square 1.
n=\frac{-1±\sqrt{1+400}}{2}
Multiply -4 times -100.
n=\frac{-1±\sqrt{401}}{2}
Add 1 to 400.
n=\frac{\sqrt{401}-1}{2}
Now solve the equation n=\frac{-1±\sqrt{401}}{2} when ± is plus. Add -1 to \sqrt{401}.
n=\frac{-\sqrt{401}-1}{2}
Now solve the equation n=\frac{-1±\sqrt{401}}{2} when ± is minus. Subtract \sqrt{401} from -1.
n=\frac{\sqrt{401}-1}{2} n=\frac{-\sqrt{401}-1}{2}
The equation is now solved.
n\left(n+1\right)=50\times 2
Multiply both sides by 2.
n^{2}+n=50\times 2
Use the distributive property to multiply n by n+1.
n^{2}+n=100
Multiply 50 and 2 to get 100.
n^{2}+n+\left(\frac{1}{2}\right)^{2}=100+\left(\frac{1}{2}\right)^{2}
Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
n^{2}+n+\frac{1}{4}=100+\frac{1}{4}
Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.
n^{2}+n+\frac{1}{4}=\frac{401}{4}
Add 100 to \frac{1}{4}.
\left(n+\frac{1}{2}\right)^{2}=\frac{401}{4}
Factor n^{2}+n+\frac{1}{4}. 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}{2}\right)^{2}}=\sqrt{\frac{401}{4}}
Take the square root of both sides of the equation.
n+\frac{1}{2}=\frac{\sqrt{401}}{2} n+\frac{1}{2}=-\frac{\sqrt{401}}{2}
Simplify.
n=\frac{\sqrt{401}-1}{2} n=\frac{-\sqrt{401}-1}{2}
Subtract \frac{1}{2} from both sides of the equation.