Solve 360/n-1+360/n+2=6

$$\frac { 360 } { n - 1 } + \frac { 360 } { n + 2 } = 6$$

Solve for n

$$n = \frac{3 \sqrt{1601} + 119}{2} \approx 119.518747071$$

$$n=\frac{119-3\sqrt{1601}}{2}\approx -0.518747071$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values -2,1 since division by zero is not defined. Multiply both sides of the equation by \left(n-1\right)\left(n+2\right), the least common multiple of n-1,n+2.

$$\left(n+2\right)\times 360+\left(n-1\right)\times 360=6\left(n-1\right)\left(n+2\right)$$

Use the distributive property to multiply n+2 by 360.

$$360n+720+\left(n-1\right)\times 360=6\left(n-1\right)\left(n+2\right)$$

Use the distributive property to multiply n-1 by 360.

$$360n+720+360n-360=6\left(n-1\right)\left(n+2\right)$$

Combine 360n and 360n to get 720n.

$$720n+720-360=6\left(n-1\right)\left(n+2\right)$$

Subtract 360 from 720 to get 360.

$$720n+360=6\left(n-1\right)\left(n+2\right)$$

Use the distributive property to multiply 6 by n-1.

$$720n+360=\left(6n-6\right)\left(n+2\right)$$

Use the distributive property to multiply 6n-6 by n+2 and combine like terms.

$$720n+360=6n^{2}+6n-12$$

Subtract 6n^{2} from both sides.

$$720n+360-6n^{2}=6n-12$$

Subtract 6n from both sides.

$$720n+360-6n^{2}-6n=-12$$

Combine 720n and -6n to get 714n.

$$714n+360-6n^{2}=-12$$

Add 12 to both sides.

$$714n+360-6n^{2}+12=0$$

Add 360 and 12 to get 372.

$$714n+372-6n^{2}=0$$

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.

$$-6n^{2}+714n+372=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 714 for b, and 372 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-714±\sqrt{714^{2}-4\left(-6\right)\times 372}}{2\left(-6\right)}$$

Square 714.

$$n=\frac{-714±\sqrt{509796-4\left(-6\right)\times 372}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$n=\frac{-714±\sqrt{509796+24\times 372}}{2\left(-6\right)}$$

Multiply 24 times 372.

$$n=\frac{-714±\sqrt{509796+8928}}{2\left(-6\right)}$$

Add 509796 to 8928.

$$n=\frac{-714±\sqrt{518724}}{2\left(-6\right)}$$

Take the square root of 518724.

$$n=\frac{-714±18\sqrt{1601}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$n=\frac{-714±18\sqrt{1601}}{-12}$$

Now solve the equation n=\frac{-714±18\sqrt{1601}}{-12} when ± is plus. Add -714 to 18\sqrt{1601}.

$$n=\frac{18\sqrt{1601}-714}{-12}$$

Divide -714+18\sqrt{1601} by -12.

$$n=\frac{119-3\sqrt{1601}}{2}$$

Now solve the equation n=\frac{-714±18\sqrt{1601}}{-12} when ± is minus. Subtract 18\sqrt{1601} from -714.

$$n=\frac{-18\sqrt{1601}-714}{-12}$$

Divide -714-18\sqrt{1601} by -12.

$$n=\frac{3\sqrt{1601}+119}{2}$$

The equation is now solved.

$$n=\frac{119-3\sqrt{1601}}{2} n=\frac{3\sqrt{1601}+119}{2}$$

Steps for Completing the Square

Variable n cannot be equal to any of the values -2,1 since division by zero is not defined. Multiply both sides of the equation by \left(n-1\right)\left(n+2\right), the least common multiple of n-1,n+2.

$$\left(n+2\right)\times 360+\left(n-1\right)\times 360=6\left(n-1\right)\left(n+2\right)$$

Use the distributive property to multiply n+2 by 360.

$$360n+720+\left(n-1\right)\times 360=6\left(n-1\right)\left(n+2\right)$$

Use the distributive property to multiply n-1 by 360.

$$360n+720+360n-360=6\left(n-1\right)\left(n+2\right)$$

Combine 360n and 360n to get 720n.

$$720n+720-360=6\left(n-1\right)\left(n+2\right)$$

Subtract 360 from 720 to get 360.

$$720n+360=6\left(n-1\right)\left(n+2\right)$$

Use the distributive property to multiply 6 by n-1.

$$720n+360=\left(6n-6\right)\left(n+2\right)$$

Use the distributive property to multiply 6n-6 by n+2 and combine like terms.

$$720n+360=6n^{2}+6n-12$$

Subtract 6n^{2} from both sides.

$$720n+360-6n^{2}=6n-12$$

Subtract 6n from both sides.

$$720n+360-6n^{2}-6n=-12$$

Combine 720n and -6n to get 714n.

$$714n+360-6n^{2}=-12$$

Subtract 360 from both sides.

$$714n-6n^{2}=-12-360$$

Subtract 360 from -12 to get -372.

$$714n-6n^{2}=-372$$

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.

$$-6n^{2}+714n=-372$$

Divide both sides by -6.

$$\frac{-6n^{2}+714n}{-6}=-\frac{372}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$n^{2}+\frac{714}{-6}n=-\frac{372}{-6}$$

Divide 714 by -6.

$$n^{2}-119n=-\frac{372}{-6}$$

Divide -372 by -6.

$$n^{2}-119n=62$$

Divide -119, the coefficient of the x term, by 2 to get -\frac{119}{2}. Then add the square of -\frac{119}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-119n+\left(-\frac{119}{2}\right)^{2}=62+\left(-\frac{119}{2}\right)^{2}$$

Square -\frac{119}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-119n+\frac{14161}{4}=62+\frac{14161}{4}$$

Add 62 to \frac{14161}{4}.

$$n^{2}-119n+\frac{14161}{4}=\frac{14409}{4}$$

Factor n^{2}-119n+\frac{14161}{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}.

$$\left(n-\frac{119}{2}\right)^{2}=\frac{14409}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{119}{2}\right)^{2}}=\sqrt{\frac{14409}{4}}$$

Simplify.

$$n-\frac{119}{2}=\frac{3\sqrt{1601}}{2} n-\frac{119}{2}=-\frac{3\sqrt{1601}}{2}$$

Add \frac{119}{2} to both sides of the equation.

$$n=\frac{3\sqrt{1601}+119}{2} n=\frac{119-3\sqrt{1601}}{2}$$