Solve 116=n/2[50+(n-1)-3]

$$116 = \frac { n } { 2 } [ 50 + ( n - 1 ) - 3 ]$$

Solve for n

$$n=\sqrt{761}-23\approx 4.586228448$$

$$n=-\sqrt{761}-23\approx -50.586228448$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$232=n\left(50+n-1-3\right)$$

Subtract 1 from 50 to get 49.

$$232=n\left(49+n-3\right)$$

Subtract 3 from 49 to get 46.

$$232=n\left(46+n\right)$$

Use the distributive property to multiply n by 46+n.

$$232=46n+n^{2}$$

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

$$46n+n^{2}=232$$

Subtract 232 from both sides.

$$46n+n^{2}-232=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$n^{2}+46n-232=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 46 for b, and -232 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-46±\sqrt{46^{2}-4\left(-232\right)}}{2}$$

Square 46.

$$n=\frac{-46±\sqrt{2116-4\left(-232\right)}}{2}$$

Multiply -4 times -232.

$$n=\frac{-46±\sqrt{2116+928}}{2}$$

Add 2116 to 928.

$$n=\frac{-46±\sqrt{3044}}{2}$$

Take the square root of 3044.

$$n=\frac{-46±2\sqrt{761}}{2}$$

Now solve the equation n=\frac{-46±2\sqrt{761}}{2} when ± is plus. Add -46 to 2\sqrt{761}.

$$n=\frac{2\sqrt{761}-46}{2}$$

Divide -46+2\sqrt{761} by 2.

$$n=\sqrt{761}-23$$

Now solve the equation n=\frac{-46±2\sqrt{761}}{2} when ± is minus. Subtract 2\sqrt{761} from -46.

$$n=\frac{-2\sqrt{761}-46}{2}$$

Divide -46-2\sqrt{761} by 2.

$$n=-\sqrt{761}-23$$

The equation is now solved.

$$n=\sqrt{761}-23 n=-\sqrt{761}-23$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$232=n\left(50+n-1-3\right)$$

Subtract 1 from 50 to get 49.

$$232=n\left(49+n-3\right)$$

Subtract 3 from 49 to get 46.

$$232=n\left(46+n\right)$$

Use the distributive property to multiply n by 46+n.

$$232=46n+n^{2}$$

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

$$46n+n^{2}=232$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$n^{2}+46n=232$$

Divide 46, the coefficient of the x term, by 2 to get 23. Then add the square of 23 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+46n+23^{2}=232+23^{2}$$

Square 23.

$$n^{2}+46n+529=232+529$$

Add 232 to 529.

$$n^{2}+46n+529=761$$

Factor n^{2}+46n+529. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+23\right)^{2}=761$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+23\right)^{2}}=\sqrt{761}$$

Simplify.

$$n+23=\sqrt{761} n+23=-\sqrt{761}$$

Subtract 23 from both sides of the equation.

$$n=\sqrt{761}-23 n=-\sqrt{761}-23$$