Solve 2(1000/x)+x=125

$$2 ( \frac { 1000 } { x } ) + x = 125$$

Solve for x

$$x = \frac{5 \sqrt{305} + 125}{2} \approx 106.160622991$$

$$x = \frac{125 - 5 \sqrt{305}}{2} \approx 18.839377009$$

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.

$$2\times 1000+xx=125x$$

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

$$2\times 1000+x^{2}=125x$$

Multiply 2 and 1000 to get 2000.

$$2000+x^{2}=125x$$

Subtract 125x from both sides.

$$2000+x^{2}-125x=0$$

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.

$$x^{2}-125x+2000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -125 for b, and 2000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-125\right)±\sqrt{\left(-125\right)^{2}-4\times 2000}}{2}$$

Square -125.

$$x=\frac{-\left(-125\right)±\sqrt{15625-4\times 2000}}{2}$$

Multiply -4 times 2000.

$$x=\frac{-\left(-125\right)±\sqrt{15625-8000}}{2}$$

Add 15625 to -8000.

$$x=\frac{-\left(-125\right)±\sqrt{7625}}{2}$$

Take the square root of 7625.

$$x=\frac{-\left(-125\right)±5\sqrt{305}}{2}$$

The opposite of -125 is 125.

$$x=\frac{125±5\sqrt{305}}{2}$$

Now solve the equation x=\frac{125±5\sqrt{305}}{2} when ± is plus. Add 125 to 5\sqrt{305}.

$$x=\frac{5\sqrt{305}+125}{2}$$

Now solve the equation x=\frac{125±5\sqrt{305}}{2} when ± is minus. Subtract 5\sqrt{305} from 125.

$$x=\frac{125-5\sqrt{305}}{2}$$

The equation is now solved.

$$x=\frac{5\sqrt{305}+125}{2} x=\frac{125-5\sqrt{305}}{2}$$

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.

$$2\times 1000+xx=125x$$

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

$$2\times 1000+x^{2}=125x$$

Multiply 2 and 1000 to get 2000.

$$2000+x^{2}=125x$$

Subtract 125x from both sides.

$$2000+x^{2}-125x=0$$

Subtract 2000 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-125x=-2000$$

Divide -125, the coefficient of the x term, by 2 to get -\frac{125}{2}. Then add the square of -\frac{125}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-125x+\left(-\frac{125}{2}\right)^{2}=-2000+\left(-\frac{125}{2}\right)^{2}$$

Square -\frac{125}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-125x+\frac{15625}{4}=-2000+\frac{15625}{4}$$

Add -2000 to \frac{15625}{4}.

$$x^{2}-125x+\frac{15625}{4}=\frac{7625}{4}$$

Factor x^{2}-125x+\frac{15625}{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}.

$$\left(x-\frac{125}{2}\right)^{2}=\frac{7625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{125}{2}\right)^{2}}=\sqrt{\frac{7625}{4}}$$

Simplify.

$$x-\frac{125}{2}=\frac{5\sqrt{305}}{2} x-\frac{125}{2}=-\frac{5\sqrt{305}}{2}$$

Add \frac{125}{2} to both sides of the equation.

$$x=\frac{5\sqrt{305}+125}{2} x=\frac{125-5\sqrt{305}}{2}$$