Solve 7/3n=2n+8/3n^2+1/6

$$\frac { 7 } { 3 n } = \frac { 2 n + 8 } { 3 n ^ { 2 } } + \frac { 1 } { 6 }$$

Solve for n

$$n=2$$

$$n=8$$

Steps Using Factoring By Grouping

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6n^{2}, the least common multiple of 3n,3n^{2},6.

$$2n\times 7=2\left(2n+8\right)+6n^{2}\times \frac{1}{6}$$

Multiply 2 and 7 to get 14.

$$14n=2\left(2n+8\right)+6n^{2}\times \frac{1}{6}$$

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

$$14n=4n+16+6n^{2}\times \frac{1}{6}$$

Multiply 6 and \frac{1}{6} to get 1.

$$14n=4n+16+n^{2}$$

Subtract 4n from both sides.

$$14n-4n=16+n^{2}$$

Combine 14n and -4n to get 10n.

$$10n=16+n^{2}$$

Subtract 16 from both sides.

$$10n-16=n^{2}$$

Subtract n^{2} from both sides.

$$10n-16-n^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-n^{2}+10n-16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -n^{2}+an+bn-16. To find a and b, set up a system to be solved.

$$a+b=10 ab=-\left(-16\right)=16$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 16.

$$1,16 2,8 4,4$$

Calculate the sum for each pair.

$$1+16=17 2+8=10 4+4=8$$

The solution is the pair that gives sum 10.

$$a=8 b=2$$

Rewrite -n^{2}+10n-16 as \left(-n^{2}+8n\right)+\left(2n-16\right).

$$\left(-n^{2}+8n\right)+\left(2n-16\right)$$

Factor out -n in the first and 2 in the second group.

$$-n\left(n-8\right)+2\left(n-8\right)$$

Factor out common term n-8 by using distributive property.

$$\left(n-8\right)\left(-n+2\right)$$

To find equation solutions, solve n-8=0 and -n+2=0.

$$n=8 n=2$$

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 6n^{2}, the least common multiple of 3n,3n^{2},6.

$$2n\times 7=2\left(2n+8\right)+6n^{2}\times \frac{1}{6}$$

Multiply 2 and 7 to get 14.

$$14n=2\left(2n+8\right)+6n^{2}\times \frac{1}{6}$$

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

$$14n=4n+16+6n^{2}\times \frac{1}{6}$$

Multiply 6 and \frac{1}{6} to get 1.

$$14n=4n+16+n^{2}$$

Subtract 4n from both sides.

$$14n-4n=16+n^{2}$$

Combine 14n and -4n to get 10n.

$$10n=16+n^{2}$$

Subtract 16 from both sides.

$$10n-16=n^{2}$$

Subtract n^{2} from both sides.

$$10n-16-n^{2}=0$$

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.

$$-n^{2}+10n-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 10 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Square 10.

$$n=\frac{-10±\sqrt{100-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{-10±\sqrt{100+4\left(-16\right)}}{2\left(-1\right)}$$

Multiply 4 times -16.

$$n=\frac{-10±\sqrt{100-64}}{2\left(-1\right)}$$

Add 100 to -64.

$$n=\frac{-10±\sqrt{36}}{2\left(-1\right)}$$

Take the square root of 36.

$$n=\frac{-10±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{-10±6}{-2}$$

Now solve the equation n=\frac{-10±6}{-2} when ± is plus. Add -10 to 6.

$$n=-\frac{4}{-2}$$

Divide -4 by -2.

$$n=2$$

Now solve the equation n=\frac{-10±6}{-2} when ± is minus. Subtract 6 from -10.

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

Divide -16 by -2.

$$n=8$$

The equation is now solved.

$$n=2 n=8$$

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 6n^{2}, the least common multiple of 3n,3n^{2},6.

$$2n\times 7=2\left(2n+8\right)+6n^{2}\times \frac{1}{6}$$

Multiply 2 and 7 to get 14.

$$14n=2\left(2n+8\right)+6n^{2}\times \frac{1}{6}$$

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

$$14n=4n+16+6n^{2}\times \frac{1}{6}$$

Multiply 6 and \frac{1}{6} to get 1.

$$14n=4n+16+n^{2}$$

Subtract 4n from both sides.

$$14n-4n=16+n^{2}$$

Combine 14n and -4n to get 10n.

$$10n=16+n^{2}$$

Subtract n^{2} from both sides.

$$10n-n^{2}=16$$

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.

$$-n^{2}+10n=16$$

Divide both sides by -1.

$$\frac{-n^{2}+10n}{-1}=\frac{16}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$n^{2}+\frac{10}{-1}n=\frac{16}{-1}$$

Divide 10 by -1.

$$n^{2}-10n=\frac{16}{-1}$$

Divide 16 by -1.

$$n^{2}-10n=-16$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-10n+\left(-5\right)^{2}=-16+\left(-5\right)^{2}$$

Square -5.

$$n^{2}-10n+25=-16+25$$

Add -16 to 25.

$$n^{2}-10n+25=9$$

Factor n^{2}-10n+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-5\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-5\right)^{2}}=\sqrt{9}$$

Simplify.

$$n-5=3 n-5=-3$$

Add 5 to both sides of the equation.

$$n=8 n=2$$