Solve 300/10-(frac{900{x}+2)}=x

$$\frac{ 300 }{ 10-( \frac{ 900 }{ x } +2) } =x$$

Solve for x

$$x=150$$

Steps Using Factoring

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x}{x}.

$$\frac{300}{10-\left(\frac{900}{x}+\frac{2x}{x}\right)}=x$$

Since \frac{900}{x} and \frac{2x}{x} have the same denominator, add them by adding their numerators.

$$\frac{300}{10-\frac{900+2x}{x}}=x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 10 times \frac{x}{x}.

$$\frac{300}{\frac{10x}{x}-\frac{900+2x}{x}}=x$$

Since \frac{10x}{x} and \frac{900+2x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{300}{\frac{10x-\left(900+2x\right)}{x}}=x$$

Do the multiplications in 10x-\left(900+2x\right).

$$\frac{300}{\frac{10x-900-2x}{x}}=x$$

Combine like terms in 10x-900-2x.

$$\frac{300}{\frac{8x-900}{x}}=x$$

Variable x cannot be equal to 0 since division by zero is not defined. Divide 300 by \frac{8x-900}{x} by multiplying 300 by the reciprocal of \frac{8x-900}{x}.

$$\frac{300x}{8x-900}=x$$

Factor the expressions that are not already factored in \frac{300x}{8x-900}.

$$\frac{300x}{4\left(2x-225\right)}=x$$

Cancel out 4 in both numerator and denominator.

$$\frac{75x}{2x-225}=x$$

Subtract x from both sides.

$$\frac{75x}{2x-225}-x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2x-225}{2x-225}.

$$\frac{75x}{2x-225}-\frac{x\left(2x-225\right)}{2x-225}=0$$

Since \frac{75x}{2x-225} and \frac{x\left(2x-225\right)}{2x-225} have the same denominator, subtract them by subtracting their numerators.

$$\frac{75x-x\left(2x-225\right)}{2x-225}=0$$

Do the multiplications in 75x-x\left(2x-225\right).

$$\frac{75x-2x^{2}+225x}{2x-225}=0$$

Combine like terms in 75x-2x^{2}+225x.

$$\frac{300x-2x^{2}}{2x-225}=0$$

Variable x cannot be equal to \frac{225}{2} since division by zero is not defined. Multiply both sides of the equation by 2x-225.

$$300x-2x^{2}=0$$

Factor out x.

$$x\left(300-2x\right)=0$$

To find equation solutions, solve x=0 and 300-2x=0.

$$x=0 x=150$$

Variable x cannot be equal to 0.

$$x=150$$

Steps Using the Quadratic Formula

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x}{x}.

$$\frac{300}{10-\left(\frac{900}{x}+\frac{2x}{x}\right)}=x$$

Since \frac{900}{x} and \frac{2x}{x} have the same denominator, add them by adding their numerators.

$$\frac{300}{10-\frac{900+2x}{x}}=x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 10 times \frac{x}{x}.

$$\frac{300}{\frac{10x}{x}-\frac{900+2x}{x}}=x$$

Since \frac{10x}{x} and \frac{900+2x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{300}{\frac{10x-\left(900+2x\right)}{x}}=x$$

Do the multiplications in 10x-\left(900+2x\right).

$$\frac{300}{\frac{10x-900-2x}{x}}=x$$

Combine like terms in 10x-900-2x.

$$\frac{300}{\frac{8x-900}{x}}=x$$

Variable x cannot be equal to 0 since division by zero is not defined. Divide 300 by \frac{8x-900}{x} by multiplying 300 by the reciprocal of \frac{8x-900}{x}.

$$\frac{300x}{8x-900}=x$$

Factor the expressions that are not already factored in \frac{300x}{8x-900}.

$$\frac{300x}{4\left(2x-225\right)}=x$$

Cancel out 4 in both numerator and denominator.

$$\frac{75x}{2x-225}=x$$

Subtract x from both sides.

$$\frac{75x}{2x-225}-x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2x-225}{2x-225}.

$$\frac{75x}{2x-225}-\frac{x\left(2x-225\right)}{2x-225}=0$$

Since \frac{75x}{2x-225} and \frac{x\left(2x-225\right)}{2x-225} have the same denominator, subtract them by subtracting their numerators.

$$\frac{75x-x\left(2x-225\right)}{2x-225}=0$$

Do the multiplications in 75x-x\left(2x-225\right).

$$\frac{75x-2x^{2}+225x}{2x-225}=0$$

Combine like terms in 75x-2x^{2}+225x.

$$\frac{300x-2x^{2}}{2x-225}=0$$

Variable x cannot be equal to \frac{225}{2} since division by zero is not defined. Multiply both sides of the equation by 2x-225.

$$300x-2x^{2}=0$$

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.

$$-2x^{2}+300x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 300 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-300±\sqrt{300^{2}}}{2\left(-2\right)}$$

Take the square root of 300^{2}.

$$x=\frac{-300±300}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-300±300}{-4}$$

Now solve the equation x=\frac{-300±300}{-4} when ± is plus. Add -300 to 300.

$$x=\frac{0}{-4}$$

Divide 0 by -4.

$$x=0$$

Now solve the equation x=\frac{-300±300}{-4} when ± is minus. Subtract 300 from -300.

$$x=-\frac{600}{-4}$$

Divide -600 by -4.

$$x=150$$

The equation is now solved.

$$x=0 x=150$$

Variable x cannot be equal to 0.

$$x=150$$

Steps for Completing the Square

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x}{x}.

$$\frac{300}{10-\left(\frac{900}{x}+\frac{2x}{x}\right)}=x$$

Since \frac{900}{x} and \frac{2x}{x} have the same denominator, add them by adding their numerators.

$$\frac{300}{10-\frac{900+2x}{x}}=x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 10 times \frac{x}{x}.

$$\frac{300}{\frac{10x}{x}-\frac{900+2x}{x}}=x$$

Since \frac{10x}{x} and \frac{900+2x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{300}{\frac{10x-\left(900+2x\right)}{x}}=x$$

Do the multiplications in 10x-\left(900+2x\right).

$$\frac{300}{\frac{10x-900-2x}{x}}=x$$

Combine like terms in 10x-900-2x.

$$\frac{300}{\frac{8x-900}{x}}=x$$

Variable x cannot be equal to 0 since division by zero is not defined. Divide 300 by \frac{8x-900}{x} by multiplying 300 by the reciprocal of \frac{8x-900}{x}.

$$\frac{300x}{8x-900}=x$$

Factor the expressions that are not already factored in \frac{300x}{8x-900}.

$$\frac{300x}{4\left(2x-225\right)}=x$$

Cancel out 4 in both numerator and denominator.

$$\frac{75x}{2x-225}=x$$

Subtract x from both sides.

$$\frac{75x}{2x-225}-x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2x-225}{2x-225}.

$$\frac{75x}{2x-225}-\frac{x\left(2x-225\right)}{2x-225}=0$$

Since \frac{75x}{2x-225} and \frac{x\left(2x-225\right)}{2x-225} have the same denominator, subtract them by subtracting their numerators.

$$\frac{75x-x\left(2x-225\right)}{2x-225}=0$$

Do the multiplications in 75x-x\left(2x-225\right).

$$\frac{75x-2x^{2}+225x}{2x-225}=0$$

Combine like terms in 75x-2x^{2}+225x.

$$\frac{300x-2x^{2}}{2x-225}=0$$

Variable x cannot be equal to \frac{225}{2} since division by zero is not defined. Multiply both sides of the equation by 2x-225.

$$300x-2x^{2}=0$$

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.

$$-2x^{2}+300x=0$$

Divide both sides by -2.

$$\frac{-2x^{2}+300x}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{300}{-2}x=\frac{0}{-2}$$

Divide 300 by -2.

$$x^{2}-150x=\frac{0}{-2}$$

Divide 0 by -2.

$$x^{2}-150x=0$$

Divide -150, the coefficient of the x term, by 2 to get -75. Then add the square of -75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-150x+\left(-75\right)^{2}=\left(-75\right)^{2}$$

Square -75.

$$x^{2}-150x+5625=5625$$

Factor x^{2}-150x+5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-75\right)^{2}=5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-75\right)^{2}}=\sqrt{5625}$$

Simplify.

$$x-75=75 x-75=-75$$

Add 75 to both sides of the equation.

$$x=150 x=0$$

Variable x cannot be equal to 0.

$$x=150$$