Solve 7200(1+0.2)/x-7200/x+4=200

$$\frac { 7200 ( 1 + 0.2 ) } { x } - \frac { 7200 } { x + 4 } = 200$$

Solve for x

$$x = \frac{4 \sqrt{274} + 8}{5} \approx 14.842356286$$

$$x=\frac{8-4\sqrt{274}}{5}\approx -11.642356286$$

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.

$$\left(x+4\right)\times 7200\left(1+0.2\right)-x\times 7200=200x\left(x+4\right)$$

Add 1 and 0.2 to get 1.2.

$$\left(x+4\right)\times 7200\times 1.2-x\times 7200=200x\left(x+4\right)$$

Multiply 7200 and 1.2 to get 8640.

$$\left(x+4\right)\times 8640-x\times 7200=200x\left(x+4\right)$$

Use the distributive property to multiply x+4 by 8640.

$$8640x+34560-x\times 7200=200x\left(x+4\right)$$

Use the distributive property to multiply 200x by x+4.

$$8640x+34560-x\times 7200=200x^{2}+800x$$

Subtract 200x^{2} from both sides.

$$8640x+34560-x\times 7200-200x^{2}=800x$$

Subtract 800x from both sides.

$$8640x+34560-x\times 7200-200x^{2}-800x=0$$

Combine 8640x and -800x to get 7840x.

$$7840x+34560-x\times 7200-200x^{2}=0$$

Multiply -1 and 7200 to get -7200.

$$7840x+34560-7200x-200x^{2}=0$$

Combine 7840x and -7200x to get 640x.

$$640x+34560-200x^{2}=0$$

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.

$$-200x^{2}+640x+34560=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -200 for a, 640 for b, and 34560 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-640±\sqrt{640^{2}-4\left(-200\right)\times 34560}}{2\left(-200\right)}$$

Square 640.

$$x=\frac{-640±\sqrt{409600-4\left(-200\right)\times 34560}}{2\left(-200\right)}$$

Multiply -4 times -200.

$$x=\frac{-640±\sqrt{409600+800\times 34560}}{2\left(-200\right)}$$

Multiply 800 times 34560.

$$x=\frac{-640±\sqrt{409600+27648000}}{2\left(-200\right)}$$

Add 409600 to 27648000.

$$x=\frac{-640±\sqrt{28057600}}{2\left(-200\right)}$$

Take the square root of 28057600.

$$x=\frac{-640±320\sqrt{274}}{2\left(-200\right)}$$

Multiply 2 times -200.

$$x=\frac{-640±320\sqrt{274}}{-400}$$

Now solve the equation x=\frac{-640±320\sqrt{274}}{-400} when ± is plus. Add -640 to 320\sqrt{274}.

$$x=\frac{320\sqrt{274}-640}{-400}$$

Divide -640+320\sqrt{274} by -400.

$$x=\frac{8-4\sqrt{274}}{5}$$

Now solve the equation x=\frac{-640±320\sqrt{274}}{-400} when ± is minus. Subtract 320\sqrt{274} from -640.

$$x=\frac{-320\sqrt{274}-640}{-400}$$

Divide -640-320\sqrt{274} by -400.

$$x=\frac{4\sqrt{274}+8}{5}$$

The equation is now solved.

$$x=\frac{8-4\sqrt{274}}{5} x=\frac{4\sqrt{274}+8}{5}$$

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.

$$\left(x+4\right)\times 7200\left(1+0.2\right)-x\times 7200=200x\left(x+4\right)$$

Add 1 and 0.2 to get 1.2.

$$\left(x+4\right)\times 7200\times 1.2-x\times 7200=200x\left(x+4\right)$$

Multiply 7200 and 1.2 to get 8640.

$$\left(x+4\right)\times 8640-x\times 7200=200x\left(x+4\right)$$

Use the distributive property to multiply x+4 by 8640.

$$8640x+34560-x\times 7200=200x\left(x+4\right)$$

Use the distributive property to multiply 200x by x+4.

$$8640x+34560-x\times 7200=200x^{2}+800x$$

Subtract 200x^{2} from both sides.

$$8640x+34560-x\times 7200-200x^{2}=800x$$

Subtract 800x from both sides.

$$8640x+34560-x\times 7200-200x^{2}-800x=0$$

Combine 8640x and -800x to get 7840x.

$$7840x+34560-x\times 7200-200x^{2}=0$$

Subtract 34560 from both sides. Anything subtracted from zero gives its negation.

$$7840x-x\times 7200-200x^{2}=-34560$$

Multiply -1 and 7200 to get -7200.

$$7840x-7200x-200x^{2}=-34560$$

Combine 7840x and -7200x to get 640x.

$$640x-200x^{2}=-34560$$

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.

$$-200x^{2}+640x=-34560$$

Divide both sides by -200.

$$\frac{-200x^{2}+640x}{-200}=-\frac{34560}{-200}$$

Dividing by -200 undoes the multiplication by -200.

$$x^{2}+\frac{640}{-200}x=-\frac{34560}{-200}$$

Reduce the fraction \frac{640}{-200} to lowest terms by extracting and canceling out 40.

$$x^{2}-\frac{16}{5}x=-\frac{34560}{-200}$$

Reduce the fraction \frac{-34560}{-200} to lowest terms by extracting and canceling out 40.

$$x^{2}-\frac{16}{5}x=\frac{864}{5}$$

Divide -\frac{16}{5}, the coefficient of the x term, by 2 to get -\frac{8}{5}. Then add the square of -\frac{8}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{5}x+\left(-\frac{8}{5}\right)^{2}=\frac{864}{5}+\left(-\frac{8}{5}\right)^{2}$$

Square -\frac{8}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{5}x+\frac{64}{25}=\frac{864}{5}+\frac{64}{25}$$

Add \frac{864}{5} to \frac{64}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{16}{5}x+\frac{64}{25}=\frac{4384}{25}$$

Factor x^{2}-\frac{16}{5}x+\frac{64}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8}{5}\right)^{2}=\frac{4384}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{5}\right)^{2}}=\sqrt{\frac{4384}{25}}$$

Simplify.

$$x-\frac{8}{5}=\frac{4\sqrt{274}}{5} x-\frac{8}{5}=-\frac{4\sqrt{274}}{5}$$

Add \frac{8}{5} to both sides of the equation.

$$x=\frac{4\sqrt{274}+8}{5} x=\frac{8-4\sqrt{274}}{5}$$