Solve n(n+1)/2=289

$$\frac { n ( n + 1 ) } { 2 } = 289$$

Solve for n

$$n = \frac{3 \sqrt{257} - 1}{2} \approx 23.546829313$$

$$n=\frac{-3\sqrt{257}-1}{2}\approx -24.546829313$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$n\left(n+1\right)=289\times 2$$

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

$$n^{2}+n=289\times 2$$

Multiply 289 and 2 to get 578.

$$n^{2}+n=578$$

Subtract 578 from both sides.

$$n^{2}+n-578=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -578 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-578\right)}}{2}$$

Square 1.

$$n=\frac{-1±\sqrt{1-4\left(-578\right)}}{2}$$

Multiply -4 times -578.

$$n=\frac{-1±\sqrt{1+2312}}{2}$$

Add 1 to 2312.

$$n=\frac{-1±\sqrt{2313}}{2}$$

Take the square root of 2313.

$$n=\frac{-1±3\sqrt{257}}{2}$$

Now solve the equation n=\frac{-1±3\sqrt{257}}{2} when ± is plus. Add -1 to 3\sqrt{257}.

$$n=\frac{3\sqrt{257}-1}{2}$$

Now solve the equation n=\frac{-1±3\sqrt{257}}{2} when ± is minus. Subtract 3\sqrt{257} from -1.

$$n=\frac{-3\sqrt{257}-1}{2}$$

The equation is now solved.

$$n=\frac{3\sqrt{257}-1}{2} n=\frac{-3\sqrt{257}-1}{2}$$

Steps for Completing the Square

Multiply both sides by 2.

$$n\left(n+1\right)=289\times 2$$

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

$$n^{2}+n=289\times 2$$

Multiply 289 and 2 to get 578.

$$n^{2}+n=578$$

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.

$$n^{2}+n+\left(\frac{1}{2}\right)^{2}=578+\left(\frac{1}{2}\right)^{2}$$

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

$$n^{2}+n+\frac{1}{4}=578+\frac{1}{4}$$

Add 578 to \frac{1}{4}.

$$n^{2}+n+\frac{1}{4}=\frac{2313}{4}$$

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}.

$$\left(n+\frac{1}{2}\right)^{2}=\frac{2313}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{1}{2}\right)^{2}}=\sqrt{\frac{2313}{4}}$$

Simplify.

$$n+\frac{1}{2}=\frac{3\sqrt{257}}{2} n+\frac{1}{2}=-\frac{3\sqrt{257}}{2}$$

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

$$n=\frac{3\sqrt{257}-1}{2} n=\frac{-3\sqrt{257}-1}{2}$$