Solve 636=n/2[10+(n-1)8]

$$636 = \frac { n } { 2 } [ 10 + ( n - 1 ) 8 ]$$

Solve for n

$$n = \frac{\sqrt{10177} - 1}{8} \approx 12.485139769$$

$$n=\frac{-\sqrt{10177}-1}{8}\approx -12.735139769$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$1272=n\left(10+\left(n-1\right)\times 8\right)$$

Use the distributive property to multiply n-1 by 8.

$$1272=n\left(10+8n-8\right)$$

Subtract 8 from 10 to get 2.

$$1272=n\left(2+8n\right)$$

Use the distributive property to multiply n by 2+8n.

$$1272=2n+8n^{2}$$

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

$$2n+8n^{2}=1272$$

Subtract 1272 from both sides.

$$2n+8n^{2}-1272=0$$

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.

$$8n^{2}+2n-1272=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 2 for b, and -1272 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-2±\sqrt{2^{2}-4\times 8\left(-1272\right)}}{2\times 8}$$

Square 2.

$$n=\frac{-2±\sqrt{4-4\times 8\left(-1272\right)}}{2\times 8}$$

Multiply -4 times 8.

$$n=\frac{-2±\sqrt{4-32\left(-1272\right)}}{2\times 8}$$

Multiply -32 times -1272.

$$n=\frac{-2±\sqrt{4+40704}}{2\times 8}$$

Add 4 to 40704.

$$n=\frac{-2±\sqrt{40708}}{2\times 8}$$

Take the square root of 40708.

$$n=\frac{-2±2\sqrt{10177}}{2\times 8}$$

Multiply 2 times 8.

$$n=\frac{-2±2\sqrt{10177}}{16}$$

Now solve the equation n=\frac{-2±2\sqrt{10177}}{16} when ± is plus. Add -2 to 2\sqrt{10177}.

$$n=\frac{2\sqrt{10177}-2}{16}$$

Divide -2+2\sqrt{10177} by 16.

$$n=\frac{\sqrt{10177}-1}{8}$$

Now solve the equation n=\frac{-2±2\sqrt{10177}}{16} when ± is minus. Subtract 2\sqrt{10177} from -2.

$$n=\frac{-2\sqrt{10177}-2}{16}$$

Divide -2-2\sqrt{10177} by 16.

$$n=\frac{-\sqrt{10177}-1}{8}$$

The equation is now solved.

$$n=\frac{\sqrt{10177}-1}{8} n=\frac{-\sqrt{10177}-1}{8}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$1272=n\left(10+\left(n-1\right)\times 8\right)$$

Use the distributive property to multiply n-1 by 8.

$$1272=n\left(10+8n-8\right)$$

Subtract 8 from 10 to get 2.

$$1272=n\left(2+8n\right)$$

Use the distributive property to multiply n by 2+8n.

$$1272=2n+8n^{2}$$

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

$$2n+8n^{2}=1272$$

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.

$$8n^{2}+2n=1272$$

Divide both sides by 8.

$$\frac{8n^{2}+2n}{8}=\frac{1272}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$n^{2}+\frac{2}{8}n=\frac{1272}{8}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$n^{2}+\frac{1}{4}n=\frac{1272}{8}$$

Divide 1272 by 8.

$$n^{2}+\frac{1}{4}n=159$$

Divide \frac{1}{4}, the coefficient of the x term, by 2 to get \frac{1}{8}. Then add the square of \frac{1}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+\frac{1}{4}n+\left(\frac{1}{8}\right)^{2}=159+\left(\frac{1}{8}\right)^{2}$$

Square \frac{1}{8} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+\frac{1}{4}n+\frac{1}{64}=159+\frac{1}{64}$$

Add 159 to \frac{1}{64}.

$$n^{2}+\frac{1}{4}n+\frac{1}{64}=\frac{10177}{64}$$

Factor n^{2}+\frac{1}{4}n+\frac{1}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+\frac{1}{8}\right)^{2}=\frac{10177}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{1}{8}\right)^{2}}=\sqrt{\frac{10177}{64}}$$

Simplify.

$$n+\frac{1}{8}=\frac{\sqrt{10177}}{8} n+\frac{1}{8}=-\frac{\sqrt{10177}}{8}$$

Subtract \frac{1}{8} from both sides of the equation.

$$n=\frac{\sqrt{10177}-1}{8} n=\frac{-\sqrt{10177}-1}{8}$$