Solve n+4/5n^2=1/5+1/5n

$$\frac { n + 4 } { 5 n ^ { 2 } } = \frac { 1 } { 5 } + \frac { 1 } { 5 n }$$

Solve for n

$$n=-2$$

$$n=2$$

Steps by Finding Square Root

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5n^{2}, the least common multiple of 5n^{2},5,5n.

$$n+4=5n^{2}\times \frac{1}{5}+n$$

Multiply 5 and \frac{1}{5} to get 1.

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

Subtract n^{2} from both sides.

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

Subtract n from both sides.

$$n+4-n^{2}-n=0$$

Combine n and -n to get 0.

$$4-n^{2}=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-n^{2}=-4$$

Divide both sides by -1.

$$n^{2}=\frac{-4}{-1}$$

Fraction \frac{-4}{-1} can be simplified to 4 by removing the negative sign from both the numerator and the denominator.

$$n^{2}=4$$

Take the square root of both sides of the equation.

$$n=2 n=-2$$

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 5n^{2}, the least common multiple of 5n^{2},5,5n.

$$n+4=5n^{2}\times \frac{1}{5}+n$$

Multiply 5 and \frac{1}{5} to get 1.

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

Subtract n^{2} from both sides.

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

Subtract n from both sides.

$$n+4-n^{2}-n=0$$

Combine n and -n to get 0.

$$4-n^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-n^{2}+4=0$$

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

$$n=\frac{0±\sqrt{0^{2}-4\left(-1\right)\times 4}}{2\left(-1\right)}$$

Square 0.

$$n=\frac{0±\sqrt{-4\left(-1\right)\times 4}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{0±\sqrt{4\times 4}}{2\left(-1\right)}$$

Multiply 4 times 4.

$$n=\frac{0±\sqrt{16}}{2\left(-1\right)}$$

Take the square root of 16.

$$n=\frac{0±4}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{0±4}{-2}$$

Now solve the equation n=\frac{0±4}{-2} when ± is plus. Divide 4 by -2.

$$n=-2$$

Now solve the equation n=\frac{0±4}{-2} when ± is minus. Divide -4 by -2.

$$n=2$$

The equation is now solved.

$$n=-2 n=2$$