Solve 64+(n-1)*2/n*n=858

$$\frac { 64 + ( n - 1 ) \cdot 2 } { n } \cdot n = 858$$

Solve for n

$$n=398$$

Steps Using Factoring

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n.

$$\left(64+\left(n-1\right)\times 2\right)n=858n$$

Use the distributive property to multiply n-1 by 2.

$$\left(64+2n-2\right)n=858n$$

Subtract 2 from 64 to get 62.

$$\left(62+2n\right)n=858n$$

Use the distributive property to multiply 62+2n by n.

$$62n+2n^{2}=858n$$

Subtract 858n from both sides.

$$62n+2n^{2}-858n=0$$

Combine 62n and -858n to get -796n.

$$-796n+2n^{2}=0$$

Factor out n.

$$n\left(-796+2n\right)=0$$

To find equation solutions, solve n=0 and -796+2n=0.

$$n=0 n=398$$

Variable n cannot be equal to 0.

$$n=398$$

Steps Using the Quadratic Formula

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n.

$$\left(64+\left(n-1\right)\times 2\right)n=858n$$

Use the distributive property to multiply n-1 by 2.

$$\left(64+2n-2\right)n=858n$$

Subtract 2 from 64 to get 62.

$$\left(62+2n\right)n=858n$$

Use the distributive property to multiply 62+2n by n.

$$62n+2n^{2}=858n$$

Subtract 858n from both sides.

$$62n+2n^{2}-858n=0$$

Combine 62n and -858n to get -796n.

$$-796n+2n^{2}=0$$

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.

$$2n^{2}-796n=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -796 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-796\right)±\sqrt{\left(-796\right)^{2}}}{2\times 2}$$

Take the square root of \left(-796\right)^{2}.

$$n=\frac{-\left(-796\right)±796}{2\times 2}$$

The opposite of -796 is 796.

$$n=\frac{796±796}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{796±796}{4}$$

Now solve the equation n=\frac{796±796}{4} when ± is plus. Add 796 to 796.

$$n=\frac{1592}{4}$$

Divide 1592 by 4.

$$n=398$$

Now solve the equation n=\frac{796±796}{4} when ± is minus. Subtract 796 from 796.

$$n=\frac{0}{4}$$

Divide 0 by 4.

$$n=0$$

The equation is now solved.

$$n=398 n=0$$

Variable n cannot be equal to 0.

$$n=398$$

Steps for Completing the Square

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n.

$$\left(64+\left(n-1\right)\times 2\right)n=858n$$

Use the distributive property to multiply n-1 by 2.

$$\left(64+2n-2\right)n=858n$$

Subtract 2 from 64 to get 62.

$$\left(62+2n\right)n=858n$$

Use the distributive property to multiply 62+2n by n.

$$62n+2n^{2}=858n$$

Subtract 858n from both sides.

$$62n+2n^{2}-858n=0$$

Combine 62n and -858n to get -796n.

$$-796n+2n^{2}=0$$

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.

$$2n^{2}-796n=0$$

Divide both sides by 2.

$$\frac{2n^{2}-796n}{2}=\frac{0}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$n^{2}+\left(-\frac{796}{2}\right)n=\frac{0}{2}$$

Divide -796 by 2.

$$n^{2}-398n=\frac{0}{2}$$

Divide 0 by 2.

$$n^{2}-398n=0$$

Divide -398, the coefficient of the x term, by 2 to get -199. Then add the square of -199 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-398n+\left(-199\right)^{2}=\left(-199\right)^{2}$$

Square -199.

$$n^{2}-398n+39601=39601$$

Factor n^{2}-398n+39601. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-199\right)^{2}=39601$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-199\right)^{2}}=\sqrt{39601}$$

Simplify.

$$n-199=199 n-199=-199$$

Add 199 to both sides of the equation.

$$n=398 n=0$$

Variable n cannot be equal to 0.

$$n=398$$