Solve 72(x+10)/x+72(x+10)/x-10=144(x+10)/x+48

$$\frac { 72 ( x + 10 ) } { x } + \frac { 72 ( x + 10 ) } { x - 10 } = \frac { 144 ( x + 10 ) } { x } + 48$$

Solve for x

$$x=-5$$

$$x=30$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by x\left(x-10\right), the least common multiple of x,x-10.

$$\left(x-10\right)\times 72\left(x+10\right)+x\times 72\left(x+10\right)=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Use the distributive property to multiply x-10 by 72.

$$\left(72x-720\right)\left(x+10\right)+x\times 72\left(x+10\right)=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Use the distributive property to multiply 72x-720 by x+10 and combine like terms.

$$72x^{2}-7200+x\times 72\left(x+10\right)=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Use the distributive property to multiply x\times 72 by x+10.

$$72x^{2}-7200+72x^{2}+10x\times 72=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Multiply 10 and 72 to get 720.

$$72x^{2}-7200+72x^{2}+720x=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Combine 72x^{2} and 72x^{2} to get 144x^{2}.

$$144x^{2}-7200+720x=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Use the distributive property to multiply x-10 by 144.

$$144x^{2}-7200+720x=\left(144x-1440\right)\left(x+10\right)+x\left(x-10\right)\times 48$$

Use the distributive property to multiply 144x-1440 by x+10 and combine like terms.

$$144x^{2}-7200+720x=144x^{2}-14400+x\left(x-10\right)\times 48$$

Use the distributive property to multiply x by x-10.

$$144x^{2}-7200+720x=144x^{2}-14400+\left(x^{2}-10x\right)\times 48$$

Use the distributive property to multiply x^{2}-10x by 48.

$$144x^{2}-7200+720x=144x^{2}-14400+48x^{2}-480x$$

Combine 144x^{2} and 48x^{2} to get 192x^{2}.

$$144x^{2}-7200+720x=192x^{2}-14400-480x$$

Subtract 192x^{2} from both sides.

$$144x^{2}-7200+720x-192x^{2}=-14400-480x$$

Combine 144x^{2} and -192x^{2} to get -48x^{2}.

$$-48x^{2}-7200+720x=-14400-480x$$

Subtract -14400 from both sides.

$$-48x^{2}-7200+720x-\left(-14400\right)=-480x$$

The opposite of -14400 is 14400.

$$-48x^{2}-7200+720x+14400=-480x$$

Add 480x to both sides.

$$-48x^{2}-7200+720x+14400+480x=0$$

Add -7200 and 14400 to get 7200.

$$-48x^{2}+7200+720x+480x=0$$

Combine 720x and 480x to get 1200x.

$$-48x^{2}+7200+1200x=0$$

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.

$$-48x^{2}+1200x+7200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -48 for a, 1200 for b, and 7200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1200±\sqrt{1200^{2}-4\left(-48\right)\times 7200}}{2\left(-48\right)}$$

Square 1200.

$$x=\frac{-1200±\sqrt{1440000-4\left(-48\right)\times 7200}}{2\left(-48\right)}$$

Multiply -4 times -48.

$$x=\frac{-1200±\sqrt{1440000+192\times 7200}}{2\left(-48\right)}$$

Multiply 192 times 7200.

$$x=\frac{-1200±\sqrt{1440000+1382400}}{2\left(-48\right)}$$

Add 1440000 to 1382400.

$$x=\frac{-1200±\sqrt{2822400}}{2\left(-48\right)}$$

Take the square root of 2822400.

$$x=\frac{-1200±1680}{2\left(-48\right)}$$

Multiply 2 times -48.

$$x=\frac{-1200±1680}{-96}$$

Now solve the equation x=\frac{-1200±1680}{-96} when ± is plus. Add -1200 to 1680.

$$x=\frac{480}{-96}$$

Divide 480 by -96.

$$x=-5$$

Now solve the equation x=\frac{-1200±1680}{-96} when ± is minus. Subtract 1680 from -1200.

$$x=-\frac{2880}{-96}$$

Divide -2880 by -96.

$$x=30$$

The equation is now solved.

$$x=-5 x=30$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by x\left(x-10\right), the least common multiple of x,x-10.

$$\left(x-10\right)\times 72\left(x+10\right)+x\times 72\left(x+10\right)=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Use the distributive property to multiply x-10 by 72.

$$\left(72x-720\right)\left(x+10\right)+x\times 72\left(x+10\right)=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Use the distributive property to multiply 72x-720 by x+10 and combine like terms.

$$72x^{2}-7200+x\times 72\left(x+10\right)=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Use the distributive property to multiply x\times 72 by x+10.

$$72x^{2}-7200+72x^{2}+10x\times 72=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Multiply 10 and 72 to get 720.

$$72x^{2}-7200+72x^{2}+720x=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Combine 72x^{2} and 72x^{2} to get 144x^{2}.

$$144x^{2}-7200+720x=\left(x-10\right)\times 144\left(x+10\right)+x\left(x-10\right)\times 48$$

Use the distributive property to multiply x-10 by 144.

$$144x^{2}-7200+720x=\left(144x-1440\right)\left(x+10\right)+x\left(x-10\right)\times 48$$

Use the distributive property to multiply 144x-1440 by x+10 and combine like terms.

$$144x^{2}-7200+720x=144x^{2}-14400+x\left(x-10\right)\times 48$$

Use the distributive property to multiply x by x-10.

$$144x^{2}-7200+720x=144x^{2}-14400+\left(x^{2}-10x\right)\times 48$$

Use the distributive property to multiply x^{2}-10x by 48.

$$144x^{2}-7200+720x=144x^{2}-14400+48x^{2}-480x$$

Combine 144x^{2} and 48x^{2} to get 192x^{2}.

$$144x^{2}-7200+720x=192x^{2}-14400-480x$$

Subtract 192x^{2} from both sides.

$$144x^{2}-7200+720x-192x^{2}=-14400-480x$$

Combine 144x^{2} and -192x^{2} to get -48x^{2}.

$$-48x^{2}-7200+720x=-14400-480x$$

Add 480x to both sides.

$$-48x^{2}-7200+720x+480x=-14400$$

Combine 720x and 480x to get 1200x.

$$-48x^{2}-7200+1200x=-14400$$

Add 7200 to both sides.

$$-48x^{2}+1200x=-14400+7200$$

Add -14400 and 7200 to get -7200.

$$-48x^{2}+1200x=-7200$$

Divide both sides by -48.

$$\frac{-48x^{2}+1200x}{-48}=-\frac{7200}{-48}$$

Dividing by -48 undoes the multiplication by -48.

$$x^{2}+\frac{1200}{-48}x=-\frac{7200}{-48}$$

Divide 1200 by -48.

$$x^{2}-25x=-\frac{7200}{-48}$$

Divide -7200 by -48.

$$x^{2}-25x=150$$

Divide -25, the coefficient of the x term, by 2 to get -\frac{25}{2}. Then add the square of -\frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=150+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-25x+\frac{625}{4}=150+\frac{625}{4}$$

Add 150 to \frac{625}{4}.

$$x^{2}-25x+\frac{625}{4}=\frac{1225}{4}$$

Factor x^{2}-25x+\frac{625}{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}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{1225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{1225}{4}}$$

Simplify.

$$x-\frac{25}{2}=\frac{35}{2} x-\frac{25}{2}=-\frac{35}{2}$$

Add \frac{25}{2} to both sides of the equation.

$$x=30 x=-5$$