Solve 300=n(n+1)/2

$$300 = \frac { n ( n + 1 ) } { 2 }$$

Solve for n

$$n=-25$$

$$n=24$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$300\times 2=n\left(n+1\right)$$

Multiply 300 and 2 to get 600.

$$600=n\left(n+1\right)$$

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

$$600=n^{2}+n$$

Swap sides so that all variable terms are on the left hand side.

$$n^{2}+n=600$$

Subtract 600 from both sides.

$$n^{2}+n-600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-600\right)}}{2}$$

Square 1.

$$n=\frac{-1±\sqrt{1-4\left(-600\right)}}{2}$$

Multiply -4 times -600.

$$n=\frac{-1±\sqrt{1+2400}}{2}$$

Add 1 to 2400.

$$n=\frac{-1±\sqrt{2401}}{2}$$

Take the square root of 2401.

$$n=\frac{-1±49}{2}$$

Now solve the equation n=\frac{-1±49}{2} when ± is plus. Add -1 to 49.

$$n=\frac{48}{2}$$

Divide 48 by 2.

$$n=24$$

Now solve the equation n=\frac{-1±49}{2} when ± is minus. Subtract 49 from -1.

$$n=-\frac{50}{2}$$

Divide -50 by 2.

$$n=-25$$

The equation is now solved.

$$n=24 n=-25$$

Steps for Completing the Square

Multiply both sides by 2.

$$300\times 2=n\left(n+1\right)$$

Multiply 300 and 2 to get 600.

$$600=n\left(n+1\right)$$

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

$$600=n^{2}+n$$

Swap sides so that all variable terms are on the left hand side.

$$n^{2}+n=600$$

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.

$$n^{2}+n+\left(\frac{1}{2}\right)^{2}=600+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+n+\frac{1}{4}=600+\frac{1}{4}$$

Add 600 to \frac{1}{4}.

$$n^{2}+n+\frac{1}{4}=\frac{2401}{4}$$

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}.

$$\left(n+\frac{1}{2}\right)^{2}=\frac{2401}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{1}{2}\right)^{2}}=\sqrt{\frac{2401}{4}}$$

Simplify.

$$n+\frac{1}{2}=\frac{49}{2} n+\frac{1}{2}=-\frac{49}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$n=24 n=-25$$