Solve 1/n+9=n/3n

$$\frac { 1 } { n + 9 } = \frac { n } { 3 n }$$

Solve for n

$$n=-6$$

Steps Using Factoring

Variable n cannot be equal to any of the values -9,0 since division by zero is not defined. Multiply both sides of the equation by 3n\left(n+9\right), the least common multiple of n+9,3n.

$$3n=\left(n+9\right)n$$

Use the distributive property to multiply n+9 by n.

$$3n=n^{2}+9n$$

Subtract n^{2} from both sides.

$$3n-n^{2}=9n$$

Subtract 9n from both sides.

$$3n-n^{2}-9n=0$$

Combine 3n and -9n to get -6n.

$$-6n-n^{2}=0$$

Factor out n.

$$n\left(-6-n\right)=0$$

To find equation solutions, solve n=0 and -6-n=0.

$$n=0 n=-6$$

Variable n cannot be equal to 0.

$$n=-6$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values -9,0 since division by zero is not defined. Multiply both sides of the equation by 3n\left(n+9\right), the least common multiple of n+9,3n.

$$3n=\left(n+9\right)n$$

Use the distributive property to multiply n+9 by n.

$$3n=n^{2}+9n$$

Subtract n^{2} from both sides.

$$3n-n^{2}=9n$$

Subtract 9n from both sides.

$$3n-n^{2}-9n=0$$

Combine 3n and -9n to get -6n.

$$-6n-n^{2}=0$$

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.

$$-n^{2}-6n=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -6 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}}}{2\left(-1\right)}$$

Take the square root of \left(-6\right)^{2}.

$$n=\frac{-\left(-6\right)±6}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$n=\frac{6±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{6±6}{-2}$$

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

$$n=\frac{12}{-2}$$

Divide 12 by -2.

$$n=-6$$

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

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

Divide 0 by -2.

$$n=0$$

The equation is now solved.

$$n=-6 n=0$$

Variable n cannot be equal to 0.

$$n=-6$$

Steps for Completing the Square

Variable n cannot be equal to any of the values -9,0 since division by zero is not defined. Multiply both sides of the equation by 3n\left(n+9\right), the least common multiple of n+9,3n.

$$3n=\left(n+9\right)n$$

Use the distributive property to multiply n+9 by n.

$$3n=n^{2}+9n$$

Subtract n^{2} from both sides.

$$3n-n^{2}=9n$$

Subtract 9n from both sides.

$$3n-n^{2}-9n=0$$

Combine 3n and -9n to get -6n.

$$-6n-n^{2}=0$$

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.

$$-n^{2}-6n=0$$

Divide both sides by -1.

$$\frac{-n^{2}-6n}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$n^{2}+\left(-\frac{6}{-1}\right)n=\frac{0}{-1}$$

Divide -6 by -1.

$$n^{2}+6n=\frac{0}{-1}$$

Divide 0 by -1.

$$n^{2}+6n=0$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+6n+3^{2}=3^{2}$$

Square 3.

$$n^{2}+6n+9=9$$

Factor n^{2}+6n+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+3\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+3\right)^{2}}=\sqrt{9}$$

Simplify.

$$n+3=3 n+3=-3$$

Subtract 3 from both sides of the equation.

$$n=0 n=-6$$

Variable n cannot be equal to 0.

$$n=-6$$