Solve 720/x+6-180/x=20

$$\frac{ 720 }{ x+6 } - \frac{ 180 }{ x } =20$$

Solve for x

$$x=3$$

$$x=18$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+6\right), the least common multiple of x+6,x.

$$x\times 720-\left(x+6\right)\times 180=20x\left(x+6\right)$$

Use the distributive property to multiply x+6 by 180.

$$x\times 720-\left(180x+1080\right)=20x\left(x+6\right)$$

To find the opposite of 180x+1080, find the opposite of each term.

$$x\times 720-180x-1080=20x\left(x+6\right)$$

Combine x\times 720 and -180x to get 540x.

$$540x-1080=20x\left(x+6\right)$$

Use the distributive property to multiply 20x by x+6.

$$540x-1080=20x^{2}+120x$$

Subtract 20x^{2} from both sides.

$$540x-1080-20x^{2}=120x$$

Subtract 120x from both sides.

$$540x-1080-20x^{2}-120x=0$$

Combine 540x and -120x to get 420x.

$$420x-1080-20x^{2}=0$$

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.

$$-20x^{2}+420x-1080=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 420 for b, and -1080 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-420±\sqrt{420^{2}-4\left(-20\right)\left(-1080\right)}}{2\left(-20\right)}$$

Square 420.

$$x=\frac{-420±\sqrt{176400-4\left(-20\right)\left(-1080\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-420±\sqrt{176400+80\left(-1080\right)}}{2\left(-20\right)}$$

Multiply 80 times -1080.

$$x=\frac{-420±\sqrt{176400-86400}}{2\left(-20\right)}$$

Add 176400 to -86400.

$$x=\frac{-420±\sqrt{90000}}{2\left(-20\right)}$$

Take the square root of 90000.

$$x=\frac{-420±300}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-420±300}{-40}$$

Now solve the equation x=\frac{-420±300}{-40} when ± is plus. Add -420 to 300.

$$x=-\frac{120}{-40}$$

Divide -120 by -40.

$$x=3$$

Now solve the equation x=\frac{-420±300}{-40} when ± is minus. Subtract 300 from -420.

$$x=-\frac{720}{-40}$$

Divide -720 by -40.

$$x=18$$

The equation is now solved.

$$x=3 x=18$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -6,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+6\right), the least common multiple of x+6,x.

$$x\times 720-\left(x+6\right)\times 180=20x\left(x+6\right)$$

Use the distributive property to multiply x+6 by 180.

$$x\times 720-\left(180x+1080\right)=20x\left(x+6\right)$$

To find the opposite of 180x+1080, find the opposite of each term.

$$x\times 720-180x-1080=20x\left(x+6\right)$$

Combine x\times 720 and -180x to get 540x.

$$540x-1080=20x\left(x+6\right)$$

Use the distributive property to multiply 20x by x+6.

$$540x-1080=20x^{2}+120x$$

Subtract 20x^{2} from both sides.

$$540x-1080-20x^{2}=120x$$

Subtract 120x from both sides.

$$540x-1080-20x^{2}-120x=0$$

Combine 540x and -120x to get 420x.

$$420x-1080-20x^{2}=0$$

Add 1080 to both sides. Anything plus zero gives itself.

$$420x-20x^{2}=1080$$

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.

$$-20x^{2}+420x=1080$$

Divide both sides by -20.

$$\frac{-20x^{2}+420x}{-20}=\frac{1080}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{420}{-20}x=\frac{1080}{-20}$$

Divide 420 by -20.

$$x^{2}-21x=\frac{1080}{-20}$$

Divide 1080 by -20.

$$x^{2}-21x=-54$$

Divide -21, the coefficient of the x term, by 2 to get -\frac{21}{2}. Then add the square of -\frac{21}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-21x+\left(-\frac{21}{2}\right)^{2}=-54+\left(-\frac{21}{2}\right)^{2}$$

Square -\frac{21}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-21x+\frac{441}{4}=-54+\frac{441}{4}$$

Add -54 to \frac{441}{4}.

$$x^{2}-21x+\frac{441}{4}=\frac{225}{4}$$

Factor x^{2}-21x+\frac{441}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$x-\frac{21}{2}=\frac{15}{2} x-\frac{21}{2}=-\frac{15}{2}$$

Add \frac{21}{2} to both sides of the equation.

$$x=18 x=3$$