Solve x+2x+2/x=14000

$$x + 2 x + \frac { 2 } { x } = 14000$$

Solve for x

$$x = \frac{\sqrt{48999994} + 7000}{3} \approx 4666.66652381$$

$$x=\frac{7000-\sqrt{48999994}}{3}\approx 0.000142857$$

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 x.

$$xx+2xx+2=14000x$$

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

$$x^{2}+2xx+2=14000x$$

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

$$x^{2}+2x^{2}+2=14000x$$

Combine x^{2} and 2x^{2} to get 3x^{2}.

$$3x^{2}+2=14000x$$

Subtract 14000x from both sides.

$$3x^{2}+2-14000x=0$$

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.

$$3x^{2}-14000x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -14000 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14000\right)±\sqrt{\left(-14000\right)^{2}-4\times 3\times 2}}{2\times 3}$$

Square -14000.

$$x=\frac{-\left(-14000\right)±\sqrt{196000000-4\times 3\times 2}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-14000\right)±\sqrt{196000000-12\times 2}}{2\times 3}$$

Multiply -12 times 2.

$$x=\frac{-\left(-14000\right)±\sqrt{196000000-24}}{2\times 3}$$

Add 196000000 to -24.

$$x=\frac{-\left(-14000\right)±\sqrt{195999976}}{2\times 3}$$

Take the square root of 195999976.

$$x=\frac{-\left(-14000\right)±2\sqrt{48999994}}{2\times 3}$$

The opposite of -14000 is 14000.

$$x=\frac{14000±2\sqrt{48999994}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{14000±2\sqrt{48999994}}{6}$$

Now solve the equation x=\frac{14000±2\sqrt{48999994}}{6} when ± is plus. Add 14000 to 2\sqrt{48999994}.

$$x=\frac{2\sqrt{48999994}+14000}{6}$$

Divide 14000+2\sqrt{48999994} by 6.

$$x=\frac{\sqrt{48999994}+7000}{3}$$

Now solve the equation x=\frac{14000±2\sqrt{48999994}}{6} when ± is minus. Subtract 2\sqrt{48999994} from 14000.

$$x=\frac{14000-2\sqrt{48999994}}{6}$$

Divide 14000-2\sqrt{48999994} by 6.

$$x=\frac{7000-\sqrt{48999994}}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{48999994}+7000}{3} x=\frac{7000-\sqrt{48999994}}{3}$$

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 x.

$$xx+2xx+2=14000x$$

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

$$x^{2}+2xx+2=14000x$$

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

$$x^{2}+2x^{2}+2=14000x$$

Combine x^{2} and 2x^{2} to get 3x^{2}.

$$3x^{2}+2=14000x$$

Subtract 14000x from both sides.

$$3x^{2}+2-14000x=0$$

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}-14000x=-2$$

Divide both sides by 3.

$$\frac{3x^{2}-14000x}{3}=-\frac{2}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{14000}{3}x=-\frac{2}{3}$$

Divide -\frac{14000}{3}, the coefficient of the x term, by 2 to get -\frac{7000}{3}. Then add the square of -\frac{7000}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{14000}{3}x+\left(-\frac{7000}{3}\right)^{2}=-\frac{2}{3}+\left(-\frac{7000}{3}\right)^{2}$$

Square -\frac{7000}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{14000}{3}x+\frac{49000000}{9}=-\frac{2}{3}+\frac{49000000}{9}$$

Add -\frac{2}{3} to \frac{49000000}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{14000}{3}x+\frac{49000000}{9}=\frac{48999994}{9}$$

Factor x^{2}-\frac{14000}{3}x+\frac{49000000}{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}.

$$\left(x-\frac{7000}{3}\right)^{2}=\frac{48999994}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7000}{3}\right)^{2}}=\sqrt{\frac{48999994}{9}}$$

Simplify.

$$x-\frac{7000}{3}=\frac{\sqrt{48999994}}{3} x-\frac{7000}{3}=-\frac{\sqrt{48999994}}{3}$$

Add \frac{7000}{3} to both sides of the equation.

$$x=\frac{\sqrt{48999994}+7000}{3} x=\frac{7000-\sqrt{48999994}}{3}$$