Solve 2/x^2-35000/x+100000000=0

$$\frac { 2 } { x ^ { 2 } } - \frac { 35000 } { x } + 100000000 = 0$$

Solve for x

$$x=\frac{\sqrt{17}+7}{40000}\approx 0.000278078$$

$$x=\frac{7-\sqrt{17}}{40000}\approx 0.000071922$$

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}, the least common multiple of x^{2},x.

$$2-x\times 35000+x^{2}\times 100000000=0$$

Multiply -1 and 35000 to get -35000.

$$2-35000x+x^{2}\times 100000000=0$$

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.

$$100000000x^{2}-35000x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100000000 for a, -35000 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-35000\right)±\sqrt{\left(-35000\right)^{2}-4\times 100000000\times 2}}{2\times 100000000}$$

Square -35000.

$$x=\frac{-\left(-35000\right)±\sqrt{1225000000-4\times 100000000\times 2}}{2\times 100000000}$$

Multiply -4 times 100000000.

$$x=\frac{-\left(-35000\right)±\sqrt{1225000000-400000000\times 2}}{2\times 100000000}$$

Multiply -400000000 times 2.

$$x=\frac{-\left(-35000\right)±\sqrt{1225000000-800000000}}{2\times 100000000}$$

Add 1225000000 to -800000000.

$$x=\frac{-\left(-35000\right)±\sqrt{425000000}}{2\times 100000000}$$

Take the square root of 425000000.

$$x=\frac{-\left(-35000\right)±5000\sqrt{17}}{2\times 100000000}$$

The opposite of -35000 is 35000.

$$x=\frac{35000±5000\sqrt{17}}{2\times 100000000}$$

Multiply 2 times 100000000.

$$x=\frac{35000±5000\sqrt{17}}{200000000}$$

Now solve the equation x=\frac{35000±5000\sqrt{17}}{200000000} when ± is plus. Add 35000 to 5000\sqrt{17}.

$$x=\frac{5000\sqrt{17}+35000}{200000000}$$

Divide 35000+5000\sqrt{17} by 200000000.

$$x=\frac{\sqrt{17}+7}{40000}$$

Now solve the equation x=\frac{35000±5000\sqrt{17}}{200000000} when ± is minus. Subtract 5000\sqrt{17} from 35000.

$$x=\frac{35000-5000\sqrt{17}}{200000000}$$

Divide 35000-5000\sqrt{17} by 200000000.

$$x=\frac{7-\sqrt{17}}{40000}$$

The equation is now solved.

$$x=\frac{\sqrt{17}+7}{40000} x=\frac{7-\sqrt{17}}{40000}$$

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}, the least common multiple of x^{2},x.

$$2-x\times 35000+x^{2}\times 100000000=0$$

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

$$-x\times 35000+x^{2}\times 100000000=-2$$

Multiply -1 and 35000 to get -35000.

$$-35000x+x^{2}\times 100000000=-2$$

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.

$$100000000x^{2}-35000x=-2$$

Divide both sides by 100000000.

$$\frac{100000000x^{2}-35000x}{100000000}=-\frac{2}{100000000}$$

Dividing by 100000000 undoes the multiplication by 100000000.

$$x^{2}+\left(-\frac{35000}{100000000}\right)x=-\frac{2}{100000000}$$

Reduce the fraction \frac{-35000}{100000000} to lowest terms by extracting and canceling out 5000.

$$x^{2}-\frac{7}{20000}x=-\frac{2}{100000000}$$

Reduce the fraction \frac{-2}{100000000} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{20000}x=-\frac{1}{50000000}$$

Divide -\frac{7}{20000}, the coefficient of the x term, by 2 to get -\frac{7}{40000}. Then add the square of -\frac{7}{40000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{20000}x+\left(-\frac{7}{40000}\right)^{2}=-\frac{1}{50000000}+\left(-\frac{7}{40000}\right)^{2}$$

Square -\frac{7}{40000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{20000}x+\frac{49}{1600000000}=-\frac{1}{50000000}+\frac{49}{1600000000}$$

Add -\frac{1}{50000000} to \frac{49}{1600000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{20000}x+\frac{49}{1600000000}=\frac{17}{1600000000}$$

Factor x^{2}-\frac{7}{20000}x+\frac{49}{1600000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{40000}\right)^{2}=\frac{17}{1600000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{40000}\right)^{2}}=\sqrt{\frac{17}{1600000000}}$$

Simplify.

$$x-\frac{7}{40000}=\frac{\sqrt{17}}{40000} x-\frac{7}{40000}=-\frac{\sqrt{17}}{40000}$$

Add \frac{7}{40000} to both sides of the equation.

$$x=\frac{\sqrt{17}+7}{40000} x=\frac{7-\sqrt{17}}{40000}$$