Solve (1+2n-1)n/2=361

$$\frac { ( 1 + 2 n - 1 ) n } { 2 } = 361$$

Solve for n

$$n=19$$

$$n=-19$$

Steps by Finding Square Root

Multiply both sides by 2.

$$\left(1+2n-1\right)n=361\times 2$$

Subtract 1 from 1 to get 0.

$$2nn=361\times 2$$

Multiply n and n to get n^{2}.

$$2n^{2}=361\times 2$$

Multiply 361 and 2 to get 722.

$$2n^{2}=722$$

Divide both sides by 2.

$$n^{2}=\frac{722}{2}$$

Divide 722 by 2 to get 361.

$$n^{2}=361$$

Take the square root of both sides of the equation.

$$n=19 n=-19$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$\left(1+2n-1\right)n=361\times 2$$

Subtract 1 from 1 to get 0.

$$2nn=361\times 2$$

Multiply n and n to get n^{2}.

$$2n^{2}=361\times 2$$

Multiply 361 and 2 to get 722.

$$2n^{2}=722$$

Subtract 722 from both sides.

$$2n^{2}-722=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 0 for b, and -722 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\times 2\left(-722\right)}}{2\times 2}$$

Square 0.

$$n=\frac{0±\sqrt{-4\times 2\left(-722\right)}}{2\times 2}$$

Multiply -4 times 2.

$$n=\frac{0±\sqrt{-8\left(-722\right)}}{2\times 2}$$

Multiply -8 times -722.

$$n=\frac{0±\sqrt{5776}}{2\times 2}$$

Take the square root of 5776.

$$n=\frac{0±76}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{0±76}{4}$$

Now solve the equation n=\frac{0±76}{4} when ± is plus. Divide 76 by 4.

$$n=19$$

Now solve the equation n=\frac{0±76}{4} when ± is minus. Divide -76 by 4.

$$n=-19$$

The equation is now solved.

$$n=19 n=-19$$