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

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

Solve for n

$$n = \frac{\sqrt{4393} - 31}{2} \approx 17.63985516$$

$$n=\frac{-\sqrt{4393}-31}{2}\approx -48.63985516$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$\left(64+\left(n-1\right)\times 2\right)n=1716$$

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

$$\left(64+2n-2\right)n=1716$$

Subtract 2 from 64 to get 62.

$$\left(62+2n\right)n=1716$$

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

$$62n+2n^{2}=1716$$

Subtract 1716 from both sides.

$$62n+2n^{2}-1716=0$$

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.

$$2n^{2}+62n-1716=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 62 for b, and -1716 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-62±\sqrt{62^{2}-4\times 2\left(-1716\right)}}{2\times 2}$$

Square 62.

$$n=\frac{-62±\sqrt{3844-4\times 2\left(-1716\right)}}{2\times 2}$$

Multiply -4 times 2.

$$n=\frac{-62±\sqrt{3844-8\left(-1716\right)}}{2\times 2}$$

Multiply -8 times -1716.

$$n=\frac{-62±\sqrt{3844+13728}}{2\times 2}$$

Add 3844 to 13728.

$$n=\frac{-62±\sqrt{17572}}{2\times 2}$$

Take the square root of 17572.

$$n=\frac{-62±2\sqrt{4393}}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{-62±2\sqrt{4393}}{4}$$

Now solve the equation n=\frac{-62±2\sqrt{4393}}{4} when ± is plus. Add -62 to 2\sqrt{4393}.

$$n=\frac{2\sqrt{4393}-62}{4}$$

Divide -62+2\sqrt{4393} by 4.

$$n=\frac{\sqrt{4393}-31}{2}$$

Now solve the equation n=\frac{-62±2\sqrt{4393}}{4} when ± is minus. Subtract 2\sqrt{4393} from -62.

$$n=\frac{-2\sqrt{4393}-62}{4}$$

Divide -62-2\sqrt{4393} by 4.

$$n=\frac{-\sqrt{4393}-31}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{4393}-31}{2} n=\frac{-\sqrt{4393}-31}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$\left(64+\left(n-1\right)\times 2\right)n=1716$$

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

$$\left(64+2n-2\right)n=1716$$

Subtract 2 from 64 to get 62.

$$\left(62+2n\right)n=1716$$

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

$$62n+2n^{2}=1716$$

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.

$$2n^{2}+62n=1716$$

Divide both sides by 2.

$$\frac{2n^{2}+62n}{2}=\frac{1716}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$n^{2}+\frac{62}{2}n=\frac{1716}{2}$$

Divide 62 by 2.

$$n^{2}+31n=\frac{1716}{2}$$

Divide 1716 by 2.

$$n^{2}+31n=858$$

Divide 31, the coefficient of the x term, by 2 to get \frac{31}{2}. Then add the square of \frac{31}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+31n+\left(\frac{31}{2}\right)^{2}=858+\left(\frac{31}{2}\right)^{2}$$

Square \frac{31}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+31n+\frac{961}{4}=858+\frac{961}{4}$$

Add 858 to \frac{961}{4}.

$$n^{2}+31n+\frac{961}{4}=\frac{4393}{4}$$

Factor n^{2}+31n+\frac{961}{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}.

$$\left(n+\frac{31}{2}\right)^{2}=\frac{4393}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{31}{2}\right)^{2}}=\sqrt{\frac{4393}{4}}$$

Simplify.

$$n+\frac{31}{2}=\frac{\sqrt{4393}}{2} n+\frac{31}{2}=-\frac{\sqrt{4393}}{2}$$

Subtract \frac{31}{2} from both sides of the equation.

$$n=\frac{\sqrt{4393}-31}{2} n=\frac{-\sqrt{4393}-31}{2}$$