Solve x+4left(2+4/xright)/2x=2.5

$$\frac{ x+4 \left( 2+ \frac{ 4 }{ x } \right) }{ 2x } = 2.5$$

Solve for x

$$x=\sqrt{5}+1\approx 3.236067977$$

$$x=1-\sqrt{5}\approx -1.236067977$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x.

$$x+4\left(2+\frac{4}{x}\right)=5x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x}{x}.

$$x+4\left(\frac{2x}{x}+\frac{4}{x}\right)=5x$$

Since \frac{2x}{x} and \frac{4}{x} have the same denominator, add them by adding their numerators.

$$x+4\times \frac{2x+4}{x}=5x$$

Express 4\times \frac{2x+4}{x} as a single fraction.

$$x+\frac{4\left(2x+4\right)}{x}=5x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.

$$\frac{xx}{x}+\frac{4\left(2x+4\right)}{x}=5x$$

Since \frac{xx}{x} and \frac{4\left(2x+4\right)}{x} have the same denominator, add them by adding their numerators.

$$\frac{xx+4\left(2x+4\right)}{x}=5x$$

Do the multiplications in xx+4\left(2x+4\right).

$$\frac{x^{2}+8x+16}{x}=5x$$

Subtract 5x from both sides.

$$\frac{x^{2}+8x+16}{x}-5x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -5x times \frac{x}{x}.

$$\frac{x^{2}+8x+16}{x}+\frac{-5xx}{x}=0$$

Since \frac{x^{2}+8x+16}{x} and \frac{-5xx}{x} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+8x+16-5xx}{x}=0$$

Do the multiplications in x^{2}+8x+16-5xx.

$$\frac{x^{2}+8x+16-5x^{2}}{x}=0$$

Combine like terms in x^{2}+8x+16-5x^{2}.

$$\frac{-4x^{2}+8x+16}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-4x^{2}+8x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 8 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-4\right)\times 16}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-8±\sqrt{64+16\times 16}}{2\left(-4\right)}$$

Multiply 16 times 16.

$$x=\frac{-8±\sqrt{64+256}}{2\left(-4\right)}$$

Add 64 to 256.

$$x=\frac{-8±\sqrt{320}}{2\left(-4\right)}$$

Take the square root of 320.

$$x=\frac{-8±8\sqrt{5}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-8±8\sqrt{5}}{-8}$$

Now solve the equation x=\frac{-8±8\sqrt{5}}{-8} when ± is plus. Add -8 to 8\sqrt{5}.

$$x=\frac{8\sqrt{5}-8}{-8}$$

Divide -8+8\sqrt{5} by -8.

$$x=1-\sqrt{5}$$

Now solve the equation x=\frac{-8±8\sqrt{5}}{-8} when ± is minus. Subtract 8\sqrt{5} from -8.

$$x=\frac{-8\sqrt{5}-8}{-8}$$

Divide -8-8\sqrt{5} by -8.

$$x=\sqrt{5}+1$$

The equation is now solved.

$$x=1-\sqrt{5} x=\sqrt{5}+1$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x.

$$x+4\left(2+\frac{4}{x}\right)=5x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x}{x}.

$$x+4\left(\frac{2x}{x}+\frac{4}{x}\right)=5x$$

Since \frac{2x}{x} and \frac{4}{x} have the same denominator, add them by adding their numerators.

$$x+4\times \frac{2x+4}{x}=5x$$

Express 4\times \frac{2x+4}{x} as a single fraction.

$$x+\frac{4\left(2x+4\right)}{x}=5x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.

$$\frac{xx}{x}+\frac{4\left(2x+4\right)}{x}=5x$$

Since \frac{xx}{x} and \frac{4\left(2x+4\right)}{x} have the same denominator, add them by adding their numerators.

$$\frac{xx+4\left(2x+4\right)}{x}=5x$$

Do the multiplications in xx+4\left(2x+4\right).

$$\frac{x^{2}+8x+16}{x}=5x$$

Subtract 5x from both sides.

$$\frac{x^{2}+8x+16}{x}-5x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -5x times \frac{x}{x}.

$$\frac{x^{2}+8x+16}{x}+\frac{-5xx}{x}=0$$

Since \frac{x^{2}+8x+16}{x} and \frac{-5xx}{x} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+8x+16-5xx}{x}=0$$

Do the multiplications in x^{2}+8x+16-5xx.

$$\frac{x^{2}+8x+16-5x^{2}}{x}=0$$

Combine like terms in x^{2}+8x+16-5x^{2}.

$$\frac{-4x^{2}+8x+16}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-4x^{2}+8x+16=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$-4x^{2}+8x=-16$$

Divide both sides by -4.

$$\frac{-4x^{2}+8x}{-4}=-\frac{16}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{8}{-4}x=-\frac{16}{-4}$$

Divide 8 by -4.

$$x^{2}-2x=-\frac{16}{-4}$$

Divide -16 by -4.

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

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.

$$x^{2}-2x+1=4+1$$

Add 4 to 1.

$$x^{2}-2x+1=5$$

Factor x^{2}-2x+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}.

$$\left(x-1\right)^{2}=5$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{5}$$

Simplify.

$$x-1=\sqrt{5} x-1=-\sqrt{5}$$

Add 1 to both sides of the equation.

$$x=\sqrt{5}+1 x=1-\sqrt{5}$$