Solve 1256000+0.0027/1+x=2000000x

$$1256000+ \frac{ 0.0027 }{ 1+x } =2000000x$$

Solve for x

$$x=\frac{\sqrt{26503840054}}{200000}-0.186\approx 0.628000001$$

$$x=-\frac{\sqrt{26503840054}}{200000}-0.186\approx -1.000000001$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$\left(x+1\right)\times 1256000+0.0027=2000000x\left(x+1\right)$$

Use the distributive property to multiply x+1 by 1256000.

$$1256000x+1256000+0.0027=2000000x\left(x+1\right)$$

Add 1256000 and 0.0027 to get 1256000.0027.

$$1256000x+1256000.0027=2000000x\left(x+1\right)$$

Use the distributive property to multiply 2000000x by x+1.

$$1256000x+1256000.0027=2000000x^{2}+2000000x$$

Subtract 2000000x^{2} from both sides.

$$1256000x+1256000.0027-2000000x^{2}=2000000x$$

Subtract 2000000x from both sides.

$$1256000x+1256000.0027-2000000x^{2}-2000000x=0$$

Combine 1256000x and -2000000x to get -744000x.

$$-744000x+1256000.0027-2000000x^{2}=0$$

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.

$$-2000000x^{2}-744000x+1256000.0027=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2000000 for a, -744000 for b, and 1256000.0027 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-744000\right)±\sqrt{\left(-744000\right)^{2}-4\left(-2000000\right)\times 1256000.0027}}{2\left(-2000000\right)}$$

Square -744000.

$$x=\frac{-\left(-744000\right)±\sqrt{553536000000-4\left(-2000000\right)\times 1256000.0027}}{2\left(-2000000\right)}$$

Multiply -4 times -2000000.

$$x=\frac{-\left(-744000\right)±\sqrt{553536000000+8000000\times 1256000.0027}}{2\left(-2000000\right)}$$

Multiply 8000000 times 1256000.0027.

$$x=\frac{-\left(-744000\right)±\sqrt{553536000000+10048000021600}}{2\left(-2000000\right)}$$

Add 553536000000 to 10048000021600.

$$x=\frac{-\left(-744000\right)±\sqrt{10601536021600}}{2\left(-2000000\right)}$$

Take the square root of 10601536021600.

$$x=\frac{-\left(-744000\right)±20\sqrt{26503840054}}{2\left(-2000000\right)}$$

The opposite of -744000 is 744000.

$$x=\frac{744000±20\sqrt{26503840054}}{2\left(-2000000\right)}$$

Multiply 2 times -2000000.

$$x=\frac{744000±20\sqrt{26503840054}}{-4000000}$$

Now solve the equation x=\frac{744000±20\sqrt{26503840054}}{-4000000} when ± is plus. Add 744000 to 20\sqrt{26503840054}.

$$x=\frac{20\sqrt{26503840054}+744000}{-4000000}$$

Divide 744000+20\sqrt{26503840054} by -4000000.

$$x=-\frac{\sqrt{26503840054}}{200000}-\frac{93}{500}$$

Now solve the equation x=\frac{744000±20\sqrt{26503840054}}{-4000000} when ± is minus. Subtract 20\sqrt{26503840054} from 744000.

$$x=\frac{744000-20\sqrt{26503840054}}{-4000000}$$

Divide 744000-20\sqrt{26503840054} by -4000000.

$$x=\frac{\sqrt{26503840054}}{200000}-\frac{93}{500}$$

The equation is now solved.

$$x=-\frac{\sqrt{26503840054}}{200000}-\frac{93}{500} x=\frac{\sqrt{26503840054}}{200000}-\frac{93}{500}$$

Steps for Completing the Square

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by x+1.

$$\left(x+1\right)\times 1256000+0.0027=2000000x\left(x+1\right)$$

Use the distributive property to multiply x+1 by 1256000.

$$1256000x+1256000+0.0027=2000000x\left(x+1\right)$$

Add 1256000 and 0.0027 to get 1256000.0027.

$$1256000x+1256000.0027=2000000x\left(x+1\right)$$

Use the distributive property to multiply 2000000x by x+1.

$$1256000x+1256000.0027=2000000x^{2}+2000000x$$

Subtract 2000000x^{2} from both sides.

$$1256000x+1256000.0027-2000000x^{2}=2000000x$$

Subtract 2000000x from both sides.

$$1256000x+1256000.0027-2000000x^{2}-2000000x=0$$

Combine 1256000x and -2000000x to get -744000x.

$$-744000x+1256000.0027-2000000x^{2}=0$$

Subtract 1256000.0027 from both sides. Anything subtracted from zero gives its negation.

$$-744000x-2000000x^{2}=-1256000.0027$$

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.

$$-2000000x^{2}-744000x=-1256000.0027$$

Divide both sides by -2000000.

$$\frac{-2000000x^{2}-744000x}{-2000000}=-\frac{1256000.0027}{-2000000}$$

Dividing by -2000000 undoes the multiplication by -2000000.

$$x^{2}+\left(-\frac{744000}{-2000000}\right)x=-\frac{1256000.0027}{-2000000}$$

Reduce the fraction \frac{-744000}{-2000000} to lowest terms by extracting and canceling out 8000.

$$x^{2}+\frac{93}{250}x=-\frac{1256000.0027}{-2000000}$$

Divide -1256000.0027 by -2000000.

$$x^{2}+\frac{93}{250}x=0.62800000135$$

Divide \frac{93}{250}, the coefficient of the x term, by 2 to get \frac{93}{500}. Then add the square of \frac{93}{500} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{93}{250}x+\left(\frac{93}{500}\right)^{2}=0.62800000135+\left(\frac{93}{500}\right)^{2}$$

Square \frac{93}{500} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{93}{250}x+\frac{8649}{250000}=0.62800000135+\frac{8649}{250000}$$

Add 0.62800000135 to \frac{8649}{250000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{93}{250}x+\frac{8649}{250000}=\frac{13251920027}{20000000000}$$

Factor x^{2}+\frac{93}{250}x+\frac{8649}{250000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{93}{500}\right)^{2}=\frac{13251920027}{20000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{93}{500}\right)^{2}}=\sqrt{\frac{13251920027}{20000000000}}$$

Simplify.

$$x+\frac{93}{500}=\frac{\sqrt{26503840054}}{200000} x+\frac{93}{500}=-\frac{\sqrt{26503840054}}{200000}$$

Subtract \frac{93}{500} from both sides of the equation.

$$x=\frac{\sqrt{26503840054}}{200000}-\frac{93}{500} x=-\frac{\sqrt{26503840054}}{200000}-\frac{93}{500}$$