Solve 4800/(x-5)-4800/x=48

$$\frac { 4800 } { ( x - 5 ) } - \frac { 4800 } { x } = 48$$

Solve for x

$$x=-20$$

$$x=25$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by x\left(x-5\right), the least common multiple of x-5,x.

$$x\times 4800-\left(x-5\right)\times 4800=48x\left(x-5\right)$$

Use the distributive property to multiply x-5 by 4800.

$$x\times 4800-\left(4800x-24000\right)=48x\left(x-5\right)$$

To find the opposite of 4800x-24000, find the opposite of each term.

$$x\times 4800-4800x+24000=48x\left(x-5\right)$$

Combine x\times 4800 and -4800x to get 0.

$$24000=48x\left(x-5\right)$$

Use the distributive property to multiply 48x by x-5.

$$24000=48x^{2}-240x$$

Swap sides so that all variable terms are on the left hand side.

$$48x^{2}-240x=24000$$

Subtract 24000 from both sides.

$$48x^{2}-240x-24000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 48 for a, -240 for b, and -24000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-240\right)±\sqrt{\left(-240\right)^{2}-4\times 48\left(-24000\right)}}{2\times 48}$$

Square -240.

$$x=\frac{-\left(-240\right)±\sqrt{57600-4\times 48\left(-24000\right)}}{2\times 48}$$

Multiply -4 times 48.

$$x=\frac{-\left(-240\right)±\sqrt{57600-192\left(-24000\right)}}{2\times 48}$$

Multiply -192 times -24000.

$$x=\frac{-\left(-240\right)±\sqrt{57600+4608000}}{2\times 48}$$

Add 57600 to 4608000.

$$x=\frac{-\left(-240\right)±\sqrt{4665600}}{2\times 48}$$

Take the square root of 4665600.

$$x=\frac{-\left(-240\right)±2160}{2\times 48}$$

The opposite of -240 is 240.

$$x=\frac{240±2160}{2\times 48}$$

Multiply 2 times 48.

$$x=\frac{240±2160}{96}$$

Now solve the equation x=\frac{240±2160}{96} when ± is plus. Add 240 to 2160.

$$x=\frac{2400}{96}$$

Divide 2400 by 96.

$$x=25$$

Now solve the equation x=\frac{240±2160}{96} when ± is minus. Subtract 2160 from 240.

$$x=-\frac{1920}{96}$$

Divide -1920 by 96.

$$x=-20$$

The equation is now solved.

$$x=25 x=-20$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by x\left(x-5\right), the least common multiple of x-5,x.

$$x\times 4800-\left(x-5\right)\times 4800=48x\left(x-5\right)$$

Use the distributive property to multiply x-5 by 4800.

$$x\times 4800-\left(4800x-24000\right)=48x\left(x-5\right)$$

To find the opposite of 4800x-24000, find the opposite of each term.

$$x\times 4800-4800x+24000=48x\left(x-5\right)$$

Combine x\times 4800 and -4800x to get 0.

$$24000=48x\left(x-5\right)$$

Use the distributive property to multiply 48x by x-5.

$$24000=48x^{2}-240x$$

Swap sides so that all variable terms are on the left hand side.

$$48x^{2}-240x=24000$$

Divide both sides by 48.

$$\frac{48x^{2}-240x}{48}=\frac{24000}{48}$$

Dividing by 48 undoes the multiplication by 48.

$$x^{2}+\left(-\frac{240}{48}\right)x=\frac{24000}{48}$$

Divide -240 by 48.

$$x^{2}-5x=\frac{24000}{48}$$

Divide 24000 by 48.

$$x^{2}-5x=500$$

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

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=500+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=500+\frac{25}{4}$$

Add 500 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{2025}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{2025}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{2025}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{45}{2} x-\frac{5}{2}=-\frac{45}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=25 x=-20$$