Solve 1-n/4=1/n

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

Solve for n

$$n=2$$

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 4n, the least common multiple of 4,n.

$$4n-nn=4$$

Multiply n and n to get n^{2}.

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

Subtract 4 from both sides.

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

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.

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

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

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

Square 4.

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

Multiply -4 times -1.

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

Multiply 4 times -4.

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

Add 16 to -16.

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

Take the square root of 0.

$$n=-\frac{4}{2\left(-1\right)}$$

Multiply 2 times -1.

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

Divide -4 by -2.

$$n=2$$

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 4n, the least common multiple of 4,n.

$$4n-nn=4$$

Multiply n and n to get n^{2}.

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 4 by -1.

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

Divide 4 by -1.

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

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-4n+\left(-2\right)^{2}=-4+\left(-2\right)^{2}$$

Square -2.

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

Add -4 to 4.

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

Factor n^{2}-4n+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}.

$$\left(n-2\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-2\right)^{2}}=\sqrt{0}$$

Simplify.

$$n-2=0 n-2=0$$

Add 2 to both sides of the equation.

$$n=2 n=2$$

The equation is now solved. Solutions are the same.

$$n=2$$