Solve x/1000+x*2.5/100=x

$$\frac { x } { 1000 + x } \times \frac { 2.5 } { 100 } = x$$

Solve for x

$$x=-999.975$$

$$x=0$$

Steps Using Factoring

Variable x cannot be equal to -1000 since division by zero is not defined. Multiply both sides of the equation by 100\left(x+1000\right), the least common multiple of 1000+x,100.

$$100x\times \frac{2.5}{100}=x\times 100\left(x+1000\right)$$

Expand \frac{2.5}{100} by multiplying both numerator and the denominator by 10.

$$100x\times \frac{25}{1000}=x\times 100\left(x+1000\right)$$

Reduce the fraction \frac{25}{1000} to lowest terms by extracting and canceling out 25.

$$100x\times \frac{1}{40}=x\times 100\left(x+1000\right)$$

Multiply 100 and \frac{1}{40} to get \frac{5}{2}.

$$\frac{5}{2}x=x\times 100\left(x+1000\right)$$

Use the distributive property to multiply x\times 100 by x+1000.

$$\frac{5}{2}x=100x^{2}+1000x\times 100$$

Multiply 1000 and 100 to get 100000.

$$\frac{5}{2}x=100x^{2}+100000x$$

Subtract 100x^{2} from both sides.

$$\frac{5}{2}x-100x^{2}=100000x$$

Subtract 100000x from both sides.

$$\frac{5}{2}x-100x^{2}-100000x=0$$

Combine \frac{5}{2}x and -100000x to get -\frac{199995}{2}x.

$$-\frac{199995}{2}x-100x^{2}=0$$

Factor out x.

$$x\left(-\frac{199995}{2}-100x\right)=0$$

To find equation solutions, solve x=0 and -\frac{199995}{2}-100x=0.

$$x=0 x=-\frac{39999}{40}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -1000 since division by zero is not defined. Multiply both sides of the equation by 100\left(x+1000\right), the least common multiple of 1000+x,100.

$$100x\times \frac{2.5}{100}=x\times 100\left(x+1000\right)$$

Expand \frac{2.5}{100} by multiplying both numerator and the denominator by 10.

$$100x\times \frac{25}{1000}=x\times 100\left(x+1000\right)$$

Reduce the fraction \frac{25}{1000} to lowest terms by extracting and canceling out 25.

$$100x\times \frac{1}{40}=x\times 100\left(x+1000\right)$$

Multiply 100 and \frac{1}{40} to get \frac{5}{2}.

$$\frac{5}{2}x=x\times 100\left(x+1000\right)$$

Use the distributive property to multiply x\times 100 by x+1000.

$$\frac{5}{2}x=100x^{2}+1000x\times 100$$

Multiply 1000 and 100 to get 100000.

$$\frac{5}{2}x=100x^{2}+100000x$$

Subtract 100x^{2} from both sides.

$$\frac{5}{2}x-100x^{2}=100000x$$

Subtract 100000x from both sides.

$$\frac{5}{2}x-100x^{2}-100000x=0$$

Combine \frac{5}{2}x and -100000x to get -\frac{199995}{2}x.

$$-\frac{199995}{2}x-100x^{2}=0$$

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.

$$-100x^{2}-\frac{199995}{2}x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -100 for a, -\frac{199995}{2} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{199995}{2}\right)±\sqrt{\left(-\frac{199995}{2}\right)^{2}}}{2\left(-100\right)}$$

Take the square root of \left(-\frac{199995}{2}\right)^{2}.

$$x=\frac{-\left(-\frac{199995}{2}\right)±\frac{199995}{2}}{2\left(-100\right)}$$

The opposite of -\frac{199995}{2} is \frac{199995}{2}.

$$x=\frac{\frac{199995}{2}±\frac{199995}{2}}{2\left(-100\right)}$$

Multiply 2 times -100.

$$x=\frac{\frac{199995}{2}±\frac{199995}{2}}{-200}$$

Now solve the equation x=\frac{\frac{199995}{2}±\frac{199995}{2}}{-200} when ± is plus. Add \frac{199995}{2} to \frac{199995}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{199995}{-200}$$

Reduce the fraction \frac{199995}{-200} to lowest terms by extracting and canceling out 5.

$$x=-\frac{39999}{40}$$

Now solve the equation x=\frac{\frac{199995}{2}±\frac{199995}{2}}{-200} when ± is minus. Subtract \frac{199995}{2} from \frac{199995}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{-200}$$

Divide 0 by -200.

$$x=0$$

The equation is now solved.

$$x=-\frac{39999}{40} x=0$$

Steps for Completing the Square

Variable x cannot be equal to -1000 since division by zero is not defined. Multiply both sides of the equation by 100\left(x+1000\right), the least common multiple of 1000+x,100.

$$100x\times \frac{2.5}{100}=x\times 100\left(x+1000\right)$$

Expand \frac{2.5}{100} by multiplying both numerator and the denominator by 10.

$$100x\times \frac{25}{1000}=x\times 100\left(x+1000\right)$$

Reduce the fraction \frac{25}{1000} to lowest terms by extracting and canceling out 25.

$$100x\times \frac{1}{40}=x\times 100\left(x+1000\right)$$

Multiply 100 and \frac{1}{40} to get \frac{5}{2}.

$$\frac{5}{2}x=x\times 100\left(x+1000\right)$$

Use the distributive property to multiply x\times 100 by x+1000.

$$\frac{5}{2}x=100x^{2}+1000x\times 100$$

Multiply 1000 and 100 to get 100000.

$$\frac{5}{2}x=100x^{2}+100000x$$

Subtract 100x^{2} from both sides.

$$\frac{5}{2}x-100x^{2}=100000x$$

Subtract 100000x from both sides.

$$\frac{5}{2}x-100x^{2}-100000x=0$$

Combine \frac{5}{2}x and -100000x to get -\frac{199995}{2}x.

$$-\frac{199995}{2}x-100x^{2}=0$$

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.

$$-100x^{2}-\frac{199995}{2}x=0$$

Divide both sides by -100.

$$\frac{-100x^{2}-\frac{199995}{2}x}{-100}=\frac{0}{-100}$$

Dividing by -100 undoes the multiplication by -100.

$$x^{2}+\left(-\frac{\frac{199995}{2}}{-100}\right)x=\frac{0}{-100}$$

Divide -\frac{199995}{2} by -100.

$$x^{2}+\frac{39999}{40}x=\frac{0}{-100}$$

Divide 0 by -100.

$$x^{2}+\frac{39999}{40}x=0$$

Divide \frac{39999}{40}, the coefficient of the x term, by 2 to get \frac{39999}{80}. Then add the square of \frac{39999}{80} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{39999}{40}x+\left(\frac{39999}{80}\right)^{2}=\left(\frac{39999}{80}\right)^{2}$$

Square \frac{39999}{80} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{39999}{40}x+\frac{1599920001}{6400}=\frac{1599920001}{6400}$$

Factor x^{2}+\frac{39999}{40}x+\frac{1599920001}{6400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{39999}{80}\right)^{2}=\frac{1599920001}{6400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{39999}{80}\right)^{2}}=\sqrt{\frac{1599920001}{6400}}$$

Simplify.

$$x+\frac{39999}{80}=\frac{39999}{80} x+\frac{39999}{80}=-\frac{39999}{80}$$

Subtract \frac{39999}{80} from both sides of the equation.

$$x=0 x=-\frac{39999}{40}$$