Solve 1200=6(1200/x+1200/3x)+10(1200/x)

$$1200 = 6 ( \frac { 1200 } { x } + \frac { 1200 } { 3 x } ) + 10 ( \frac { 1200 } { x } )$$

Solve for x

$$x=18$$

Steps Using Factoring

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x, the least common multiple of x,3x.

$$3600x=6\left(\frac{1200}{x}+\frac{1200}{3x}\right)\times 3x+10\times 3\times 1200$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 3x is 3x. Multiply \frac{1200}{x} times \frac{3}{3}.

$$3600x=6\left(\frac{1200\times 3}{3x}+\frac{1200}{3x}\right)\times 3x+10\times 3\times 1200$$

Since \frac{1200\times 3}{3x} and \frac{1200}{3x} have the same denominator, add them by adding their numerators.

$$3600x=6\times \frac{1200\times 3+1200}{3x}\times 3x+10\times 3\times 1200$$

Do the multiplications in 1200\times 3+1200.

$$3600x=6\times \frac{3600+1200}{3x}\times 3x+10\times 3\times 1200$$

Do the calculations in 3600+1200.

$$3600x=6\times \frac{4800}{3x}\times 3x+10\times 3\times 1200$$

Multiply 6 and 3 to get 18.

$$3600x=18\times \frac{4800}{3x}x+10\times 3\times 1200$$

Express 18\times \frac{4800}{3x} as a single fraction.

$$3600x=\frac{18\times 4800}{3x}x+10\times 3\times 1200$$

Cancel out 3 in both numerator and denominator.

$$3600x=\frac{6\times 4800}{x}x+10\times 3\times 1200$$

Express \frac{6\times 4800}{x}x as a single fraction.

$$3600x=\frac{6\times 4800x}{x}+10\times 3\times 1200$$

Multiply 10 and 3 to get 30.

$$3600x=\frac{6\times 4800x}{x}+30\times 1200$$

Multiply 30 and 1200 to get 36000.

$$3600x=\frac{6\times 4800x}{x}+36000$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 36000 times \frac{x}{x}.

$$3600x=\frac{6\times 4800x}{x}+\frac{36000x}{x}$$

Since \frac{6\times 4800x}{x} and \frac{36000x}{x} have the same denominator, add them by adding their numerators.

$$3600x=\frac{6\times 4800x+36000x}{x}$$

Do the multiplications in 6\times 4800x+36000x.

$$3600x=\frac{28800x+36000x}{x}$$

Combine like terms in 28800x+36000x.

$$3600x=\frac{64800x}{x}$$

Subtract \frac{64800x}{x} from both sides.

$$3600x-\frac{64800x}{x}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3600x times \frac{x}{x}.

$$\frac{3600xx}{x}-\frac{64800x}{x}=0$$

Since \frac{3600xx}{x} and \frac{64800x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3600xx-64800x}{x}=0$$

Do the multiplications in 3600xx-64800x.

$$\frac{3600x^{2}-64800x}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$3600x^{2}-64800x=0$$

Factor out x.

$$x\left(3600x-64800\right)=0$$

To find equation solutions, solve x=0 and 3600x-64800=0.

$$x=0 x=18$$

Variable x cannot be equal to 0.

$$x=18$$

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 3x, the least common multiple of x,3x.

$$3600x=6\left(\frac{1200}{x}+\frac{1200}{3x}\right)\times 3x+10\times 3\times 1200$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 3x is 3x. Multiply \frac{1200}{x} times \frac{3}{3}.

$$3600x=6\left(\frac{1200\times 3}{3x}+\frac{1200}{3x}\right)\times 3x+10\times 3\times 1200$$

Since \frac{1200\times 3}{3x} and \frac{1200}{3x} have the same denominator, add them by adding their numerators.

$$3600x=6\times \frac{1200\times 3+1200}{3x}\times 3x+10\times 3\times 1200$$

Do the multiplications in 1200\times 3+1200.

$$3600x=6\times \frac{3600+1200}{3x}\times 3x+10\times 3\times 1200$$

Do the calculations in 3600+1200.

$$3600x=6\times \frac{4800}{3x}\times 3x+10\times 3\times 1200$$

Multiply 6 and 3 to get 18.

$$3600x=18\times \frac{4800}{3x}x+10\times 3\times 1200$$

Express 18\times \frac{4800}{3x} as a single fraction.

$$3600x=\frac{18\times 4800}{3x}x+10\times 3\times 1200$$

Cancel out 3 in both numerator and denominator.

$$3600x=\frac{6\times 4800}{x}x+10\times 3\times 1200$$

Express \frac{6\times 4800}{x}x as a single fraction.

$$3600x=\frac{6\times 4800x}{x}+10\times 3\times 1200$$

Multiply 10 and 3 to get 30.

$$3600x=\frac{6\times 4800x}{x}+30\times 1200$$

Multiply 30 and 1200 to get 36000.

$$3600x=\frac{6\times 4800x}{x}+36000$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 36000 times \frac{x}{x}.

$$3600x=\frac{6\times 4800x}{x}+\frac{36000x}{x}$$

Since \frac{6\times 4800x}{x} and \frac{36000x}{x} have the same denominator, add them by adding their numerators.

$$3600x=\frac{6\times 4800x+36000x}{x}$$

Do the multiplications in 6\times 4800x+36000x.

$$3600x=\frac{28800x+36000x}{x}$$

Combine like terms in 28800x+36000x.

$$3600x=\frac{64800x}{x}$$

Subtract \frac{64800x}{x} from both sides.

$$3600x-\frac{64800x}{x}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3600x times \frac{x}{x}.

$$\frac{3600xx}{x}-\frac{64800x}{x}=0$$

Since \frac{3600xx}{x} and \frac{64800x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3600xx-64800x}{x}=0$$

Do the multiplications in 3600xx-64800x.

$$\frac{3600x^{2}-64800x}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$3600x^{2}-64800x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3600 for a, -64800 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64800\right)±\sqrt{\left(-64800\right)^{2}}}{2\times 3600}$$

Take the square root of \left(-64800\right)^{2}.

$$x=\frac{-\left(-64800\right)±64800}{2\times 3600}$$

The opposite of -64800 is 64800.

$$x=\frac{64800±64800}{2\times 3600}$$

Multiply 2 times 3600.

$$x=\frac{64800±64800}{7200}$$

Now solve the equation x=\frac{64800±64800}{7200} when ± is plus. Add 64800 to 64800.

$$x=\frac{129600}{7200}$$

Divide 129600 by 7200.

$$x=18$$

Now solve the equation x=\frac{64800±64800}{7200} when ± is minus. Subtract 64800 from 64800.

$$x=\frac{0}{7200}$$

Divide 0 by 7200.

$$x=0$$

The equation is now solved.

$$x=18 x=0$$

Variable x cannot be equal to 0.

$$x=18$$

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 3x, the least common multiple of x,3x.

$$3600x=6\left(\frac{1200}{x}+\frac{1200}{3x}\right)\times 3x+10\times 3\times 1200$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 3x is 3x. Multiply \frac{1200}{x} times \frac{3}{3}.

$$3600x=6\left(\frac{1200\times 3}{3x}+\frac{1200}{3x}\right)\times 3x+10\times 3\times 1200$$

Since \frac{1200\times 3}{3x} and \frac{1200}{3x} have the same denominator, add them by adding their numerators.

$$3600x=6\times \frac{1200\times 3+1200}{3x}\times 3x+10\times 3\times 1200$$

Do the multiplications in 1200\times 3+1200.

$$3600x=6\times \frac{3600+1200}{3x}\times 3x+10\times 3\times 1200$$

Do the calculations in 3600+1200.

$$3600x=6\times \frac{4800}{3x}\times 3x+10\times 3\times 1200$$

Multiply 6 and 3 to get 18.

$$3600x=18\times \frac{4800}{3x}x+10\times 3\times 1200$$

Express 18\times \frac{4800}{3x} as a single fraction.

$$3600x=\frac{18\times 4800}{3x}x+10\times 3\times 1200$$

Cancel out 3 in both numerator and denominator.

$$3600x=\frac{6\times 4800}{x}x+10\times 3\times 1200$$

Express \frac{6\times 4800}{x}x as a single fraction.

$$3600x=\frac{6\times 4800x}{x}+10\times 3\times 1200$$

Multiply 10 and 3 to get 30.

$$3600x=\frac{6\times 4800x}{x}+30\times 1200$$

Multiply 30 and 1200 to get 36000.

$$3600x=\frac{6\times 4800x}{x}+36000$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 36000 times \frac{x}{x}.

$$3600x=\frac{6\times 4800x}{x}+\frac{36000x}{x}$$

Since \frac{6\times 4800x}{x} and \frac{36000x}{x} have the same denominator, add them by adding their numerators.

$$3600x=\frac{6\times 4800x+36000x}{x}$$

Do the multiplications in 6\times 4800x+36000x.

$$3600x=\frac{28800x+36000x}{x}$$

Combine like terms in 28800x+36000x.

$$3600x=\frac{64800x}{x}$$

Subtract \frac{64800x}{x} from both sides.

$$3600x-\frac{64800x}{x}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3600x times \frac{x}{x}.

$$\frac{3600xx}{x}-\frac{64800x}{x}=0$$

Since \frac{3600xx}{x} and \frac{64800x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3600xx-64800x}{x}=0$$

Do the multiplications in 3600xx-64800x.

$$\frac{3600x^{2}-64800x}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$3600x^{2}-64800x=0$$

Divide both sides by 3600.

$$\frac{3600x^{2}-64800x}{3600}=\frac{0}{3600}$$

Dividing by 3600 undoes the multiplication by 3600.

$$x^{2}+\left(-\frac{64800}{3600}\right)x=\frac{0}{3600}$$

Divide -64800 by 3600.

$$x^{2}-18x=\frac{0}{3600}$$

Divide 0 by 3600.

$$x^{2}-18x=0$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-18x+\left(-9\right)^{2}=\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=81$$

Factor x^{2}-18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-9\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{81}$$

Simplify.

$$x-9=9 x-9=-9$$

Add 9 to both sides of the equation.

$$x=18 x=0$$

Variable x cannot be equal to 0.

$$x=18$$