Solve 2000000divx+204x=1600000

$$2000000 \div x+204x=1600000$$

Solve for x

$$x = \frac{500 \sqrt{159898} + 200000}{51} \approx 7841.88705562$$

$$x = \frac{200000 - 500 \sqrt{159898}}{51} \approx 1.250199282$$

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.

$$2000000+204xx=1600000x$$

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

$$2000000+204x^{2}=1600000x$$

Subtract 1600000x from both sides.

$$2000000+204x^{2}-1600000x=0$$

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.

$$204x^{2}-1600000x+2000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 204 for a, -1600000 for b, and 2000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1600000\right)±\sqrt{\left(-1600000\right)^{2}-4\times 204\times 2000000}}{2\times 204}$$

Square -1600000.

$$x=\frac{-\left(-1600000\right)±\sqrt{2560000000000-4\times 204\times 2000000}}{2\times 204}$$

Multiply -4 times 204.

$$x=\frac{-\left(-1600000\right)±\sqrt{2560000000000-816\times 2000000}}{2\times 204}$$

Multiply -816 times 2000000.

$$x=\frac{-\left(-1600000\right)±\sqrt{2560000000000-1632000000}}{2\times 204}$$

Add 2560000000000 to -1632000000.

$$x=\frac{-\left(-1600000\right)±\sqrt{2558368000000}}{2\times 204}$$

Take the square root of 2558368000000.

$$x=\frac{-\left(-1600000\right)±4000\sqrt{159898}}{2\times 204}$$

The opposite of -1600000 is 1600000.

$$x=\frac{1600000±4000\sqrt{159898}}{2\times 204}$$

Multiply 2 times 204.

$$x=\frac{1600000±4000\sqrt{159898}}{408}$$

Now solve the equation x=\frac{1600000±4000\sqrt{159898}}{408} when ± is plus. Add 1600000 to 4000\sqrt{159898}.

$$x=\frac{4000\sqrt{159898}+1600000}{408}$$

Divide 1600000+4000\sqrt{159898} by 408.

$$x=\frac{500\sqrt{159898}+200000}{51}$$

Now solve the equation x=\frac{1600000±4000\sqrt{159898}}{408} when ± is minus. Subtract 4000\sqrt{159898} from 1600000.

$$x=\frac{1600000-4000\sqrt{159898}}{408}$$

Divide 1600000-4000\sqrt{159898} by 408.

$$x=\frac{200000-500\sqrt{159898}}{51}$$

The equation is now solved.

$$x=\frac{500\sqrt{159898}+200000}{51} x=\frac{200000-500\sqrt{159898}}{51}$$

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.

$$2000000+204xx=1600000x$$

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

$$2000000+204x^{2}=1600000x$$

Subtract 1600000x from both sides.

$$2000000+204x^{2}-1600000x=0$$

Subtract 2000000 from both sides. Anything subtracted from zero gives its negation.

$$204x^{2}-1600000x=-2000000$$

Divide both sides by 204.

$$\frac{204x^{2}-1600000x}{204}=-\frac{2000000}{204}$$

Dividing by 204 undoes the multiplication by 204.

$$x^{2}+\left(-\frac{1600000}{204}\right)x=-\frac{2000000}{204}$$

Reduce the fraction \frac{-1600000}{204} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{400000}{51}x=-\frac{2000000}{204}$$

Reduce the fraction \frac{-2000000}{204} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{400000}{51}x=-\frac{500000}{51}$$

Divide -\frac{400000}{51}, the coefficient of the x term, by 2 to get -\frac{200000}{51}. Then add the square of -\frac{200000}{51} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{400000}{51}x+\left(-\frac{200000}{51}\right)^{2}=-\frac{500000}{51}+\left(-\frac{200000}{51}\right)^{2}$$

Square -\frac{200000}{51} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{400000}{51}x+\frac{40000000000}{2601}=-\frac{500000}{51}+\frac{40000000000}{2601}$$

Add -\frac{500000}{51} to \frac{40000000000}{2601} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{400000}{51}x+\frac{40000000000}{2601}=\frac{39974500000}{2601}$$

Factor x^{2}-\frac{400000}{51}x+\frac{40000000000}{2601}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{200000}{51}\right)^{2}=\frac{39974500000}{2601}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{200000}{51}\right)^{2}}=\sqrt{\frac{39974500000}{2601}}$$

Simplify.

$$x-\frac{200000}{51}=\frac{500\sqrt{159898}}{51} x-\frac{200000}{51}=-\frac{500\sqrt{159898}}{51}$$

Add \frac{200000}{51} to both sides of the equation.

$$x=\frac{500\sqrt{159898}+200000}{51} x=\frac{200000-500\sqrt{159898}}{51}$$