Solve 2n+1/n=4

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

Solve for n

$$n=\frac{\sqrt{2}}{2}+1\approx 1.707106781$$

$$n=-\frac{\sqrt{2}}{2}+1\approx 0.292893219$$

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 n.

$$2nn+1=4n$$

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

$$2n^{2}+1=4n$$

Subtract 4n from both sides.

$$2n^{2}+1-4n=0$$

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.

$$2n^{2}-4n+1=0$$

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

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

Square -4.

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

Multiply -4 times 2.

$$n=\frac{-\left(-4\right)±\sqrt{16-8}}{2\times 2}$$

Add 16 to -8.

$$n=\frac{-\left(-4\right)±\sqrt{8}}{2\times 2}$$

Take the square root of 8.

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

The opposite of -4 is 4.

$$n=\frac{4±2\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{4±2\sqrt{2}}{4}$$

Now solve the equation n=\frac{4±2\sqrt{2}}{4} when ± is plus. Add 4 to 2\sqrt{2}.

$$n=\frac{2\sqrt{2}+4}{4}$$

Divide 4+2\sqrt{2} by 4.

$$n=\frac{\sqrt{2}}{2}+1$$

Now solve the equation n=\frac{4±2\sqrt{2}}{4} when ± is minus. Subtract 2\sqrt{2} from 4.

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

Divide 4-2\sqrt{2} by 4.

$$n=-\frac{\sqrt{2}}{2}+1$$

The equation is now solved.

$$n=\frac{\sqrt{2}}{2}+1 n=-\frac{\sqrt{2}}{2}+1$$

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 n.

$$2nn+1=4n$$

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

$$2n^{2}+1=4n$$

Subtract 4n from both sides.

$$2n^{2}+1-4n=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide -4 by 2.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Add -\frac{1}{2} to 1.

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

Factor n^{2}-2n+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$n-1=\frac{\sqrt{2}}{2} n-1=-\frac{\sqrt{2}}{2}$$

Add 1 to both sides of the equation.

$$n=\frac{\sqrt{2}}{2}+1 n=-\frac{\sqrt{2}}{2}+1$$