Solve 7200(1+206)/x-1200/x+4=7200

$$\frac { 7200 ( 1 + 206 ) } { x } - \frac { 1200 } { x + 4 } = 7200$$

Solve for x

$$x = \frac{\sqrt{1600321} + 1217}{12} \approx 206.836495341$$

$$x=\frac{1217-\sqrt{1600321}}{12}\approx -4.003162008$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+4\right), the least common multiple of x,x+4.

$$\left(x+4\right)\times 7200\left(1+206\right)-x\times 1200=7200x\left(x+4\right)$$

Add 1 and 206 to get 207.

$$\left(x+4\right)\times 7200\times 207-x\times 1200=7200x\left(x+4\right)$$

Multiply 7200 and 207 to get 1490400.

$$\left(x+4\right)\times 1490400-x\times 1200=7200x\left(x+4\right)$$

Use the distributive property to multiply x+4 by 1490400.

$$1490400x+5961600-x\times 1200=7200x\left(x+4\right)$$

Use the distributive property to multiply 7200x by x+4.

$$1490400x+5961600-x\times 1200=7200x^{2}+28800x$$

Subtract 7200x^{2} from both sides.

$$1490400x+5961600-x\times 1200-7200x^{2}=28800x$$

Subtract 28800x from both sides.

$$1490400x+5961600-x\times 1200-7200x^{2}-28800x=0$$

Combine 1490400x and -28800x to get 1461600x.

$$1461600x+5961600-x\times 1200-7200x^{2}=0$$

Multiply -1 and 1200 to get -1200.

$$1461600x+5961600-1200x-7200x^{2}=0$$

Combine 1461600x and -1200x to get 1460400x.

$$1460400x+5961600-7200x^{2}=0$$

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.

$$-7200x^{2}+1460400x+5961600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7200 for a, 1460400 for b, and 5961600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1460400±\sqrt{1460400^{2}-4\left(-7200\right)\times 5961600}}{2\left(-7200\right)}$$

Square 1460400.

$$x=\frac{-1460400±\sqrt{2132768160000-4\left(-7200\right)\times 5961600}}{2\left(-7200\right)}$$

Multiply -4 times -7200.

$$x=\frac{-1460400±\sqrt{2132768160000+28800\times 5961600}}{2\left(-7200\right)}$$

Multiply 28800 times 5961600.

$$x=\frac{-1460400±\sqrt{2132768160000+171694080000}}{2\left(-7200\right)}$$

Add 2132768160000 to 171694080000.

$$x=\frac{-1460400±\sqrt{2304462240000}}{2\left(-7200\right)}$$

Take the square root of 2304462240000.

$$x=\frac{-1460400±1200\sqrt{1600321}}{2\left(-7200\right)}$$

Multiply 2 times -7200.

$$x=\frac{-1460400±1200\sqrt{1600321}}{-14400}$$

Now solve the equation x=\frac{-1460400±1200\sqrt{1600321}}{-14400} when ± is plus. Add -1460400 to 1200\sqrt{1600321}.

$$x=\frac{1200\sqrt{1600321}-1460400}{-14400}$$

Divide -1460400+1200\sqrt{1600321} by -14400.

$$x=\frac{1217-\sqrt{1600321}}{12}$$

Now solve the equation x=\frac{-1460400±1200\sqrt{1600321}}{-14400} when ± is minus. Subtract 1200\sqrt{1600321} from -1460400.

$$x=\frac{-1200\sqrt{1600321}-1460400}{-14400}$$

Divide -1460400-1200\sqrt{1600321} by -14400.

$$x=\frac{\sqrt{1600321}+1217}{12}$$

The equation is now solved.

$$x=\frac{1217-\sqrt{1600321}}{12} x=\frac{\sqrt{1600321}+1217}{12}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+4\right), the least common multiple of x,x+4.

$$\left(x+4\right)\times 7200\left(1+206\right)-x\times 1200=7200x\left(x+4\right)$$

Add 1 and 206 to get 207.

$$\left(x+4\right)\times 7200\times 207-x\times 1200=7200x\left(x+4\right)$$

Multiply 7200 and 207 to get 1490400.

$$\left(x+4\right)\times 1490400-x\times 1200=7200x\left(x+4\right)$$

Use the distributive property to multiply x+4 by 1490400.

$$1490400x+5961600-x\times 1200=7200x\left(x+4\right)$$

Use the distributive property to multiply 7200x by x+4.

$$1490400x+5961600-x\times 1200=7200x^{2}+28800x$$

Subtract 7200x^{2} from both sides.

$$1490400x+5961600-x\times 1200-7200x^{2}=28800x$$

Subtract 28800x from both sides.

$$1490400x+5961600-x\times 1200-7200x^{2}-28800x=0$$

Combine 1490400x and -28800x to get 1461600x.

$$1461600x+5961600-x\times 1200-7200x^{2}=0$$

Subtract 5961600 from both sides. Anything subtracted from zero gives its negation.

$$1461600x-x\times 1200-7200x^{2}=-5961600$$

Multiply -1 and 1200 to get -1200.

$$1461600x-1200x-7200x^{2}=-5961600$$

Combine 1461600x and -1200x to get 1460400x.

$$1460400x-7200x^{2}=-5961600$$

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.

$$-7200x^{2}+1460400x=-5961600$$

Divide both sides by -7200.

$$\frac{-7200x^{2}+1460400x}{-7200}=-\frac{5961600}{-7200}$$

Dividing by -7200 undoes the multiplication by -7200.

$$x^{2}+\frac{1460400}{-7200}x=-\frac{5961600}{-7200}$$

Reduce the fraction \frac{1460400}{-7200} to lowest terms by extracting and canceling out 1200.

$$x^{2}-\frac{1217}{6}x=-\frac{5961600}{-7200}$$

Divide -5961600 by -7200.

$$x^{2}-\frac{1217}{6}x=828$$

Divide -\frac{1217}{6}, the coefficient of the x term, by 2 to get -\frac{1217}{12}. Then add the square of -\frac{1217}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1217}{6}x+\left(-\frac{1217}{12}\right)^{2}=828+\left(-\frac{1217}{12}\right)^{2}$$

Square -\frac{1217}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1217}{6}x+\frac{1481089}{144}=828+\frac{1481089}{144}$$

Add 828 to \frac{1481089}{144}.

$$x^{2}-\frac{1217}{6}x+\frac{1481089}{144}=\frac{1600321}{144}$$

Factor x^{2}-\frac{1217}{6}x+\frac{1481089}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1217}{12}\right)^{2}=\frac{1600321}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1217}{12}\right)^{2}}=\sqrt{\frac{1600321}{144}}$$

Simplify.

$$x-\frac{1217}{12}=\frac{\sqrt{1600321}}{12} x-\frac{1217}{12}=-\frac{\sqrt{1600321}}{12}$$

Add \frac{1217}{12} to both sides of the equation.

$$x=\frac{\sqrt{1600321}+1217}{12} x=\frac{1217-\sqrt{1600321}}{12}$$