Solve 800/x=800/x-30-24

$$\frac{ 800 }{ x } = \frac{ 800 }{ x-30 } -24$$

Solve for x

$$x=-20$$

$$x=50$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,30 since division by zero is not defined. Multiply both sides of the equation by x\left(x-30\right), the least common multiple of x,x-30.

$$\left(x-30\right)\times 800=x\times 800+x\left(x-30\right)\left(-24\right)$$

Use the distributive property to multiply x-30 by 800.

$$800x-24000=x\times 800+x\left(x-30\right)\left(-24\right)$$

Use the distributive property to multiply x by x-30.

$$800x-24000=x\times 800+\left(x^{2}-30x\right)\left(-24\right)$$

Use the distributive property to multiply x^{2}-30x by -24.

$$800x-24000=x\times 800-24x^{2}+720x$$

Combine x\times 800 and 720x to get 1520x.

$$800x-24000=1520x-24x^{2}$$

Subtract 1520x from both sides.

$$800x-24000-1520x=-24x^{2}$$

Combine 800x and -1520x to get -720x.

$$-720x-24000=-24x^{2}$$

Add 24x^{2} to both sides.

$$-720x-24000+24x^{2}=0$$

Divide both sides by 24.

$$-30x-1000+x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-30x-1000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-1000. To find a and b, set up a system to be solved.

$$a+b=-30 ab=1\left(-1000\right)=-1000$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -1000.

$$1,-1000 2,-500 4,-250 5,-200 8,-125 10,-100 20,-50 25,-40$$

Calculate the sum for each pair.

$$1-1000=-999 2-500=-498 4-250=-246 5-200=-195 8-125=-117 10-100=-90 20-50=-30 25-40=-15$$

The solution is the pair that gives sum -30.

$$a=-50 b=20$$

Rewrite x^{2}-30x-1000 as \left(x^{2}-50x\right)+\left(20x-1000\right).

$$\left(x^{2}-50x\right)+\left(20x-1000\right)$$

Factor out x in the first and 20 in the second group.

$$x\left(x-50\right)+20\left(x-50\right)$$

Factor out common term x-50 by using distributive property.

$$\left(x-50\right)\left(x+20\right)$$

To find equation solutions, solve x-50=0 and x+20=0.

$$x=50 x=-20$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,30 since division by zero is not defined. Multiply both sides of the equation by x\left(x-30\right), the least common multiple of x,x-30.

$$\left(x-30\right)\times 800=x\times 800+x\left(x-30\right)\left(-24\right)$$

Use the distributive property to multiply x-30 by 800.

$$800x-24000=x\times 800+x\left(x-30\right)\left(-24\right)$$

Use the distributive property to multiply x by x-30.

$$800x-24000=x\times 800+\left(x^{2}-30x\right)\left(-24\right)$$

Use the distributive property to multiply x^{2}-30x by -24.

$$800x-24000=x\times 800-24x^{2}+720x$$

Combine x\times 800 and 720x to get 1520x.

$$800x-24000=1520x-24x^{2}$$

Subtract 1520x from both sides.

$$800x-24000-1520x=-24x^{2}$$

Combine 800x and -1520x to get -720x.

$$-720x-24000=-24x^{2}$$

Add 24x^{2} to both sides.

$$-720x-24000+24x^{2}=0$$

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.

$$24x^{2}-720x-24000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, -720 for b, and -24000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-720\right)±\sqrt{\left(-720\right)^{2}-4\times 24\left(-24000\right)}}{2\times 24}$$

Square -720.

$$x=\frac{-\left(-720\right)±\sqrt{518400-4\times 24\left(-24000\right)}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-720\right)±\sqrt{518400-96\left(-24000\right)}}{2\times 24}$$

Multiply -96 times -24000.

$$x=\frac{-\left(-720\right)±\sqrt{518400+2304000}}{2\times 24}$$

Add 518400 to 2304000.

$$x=\frac{-\left(-720\right)±\sqrt{2822400}}{2\times 24}$$

Take the square root of 2822400.

$$x=\frac{-\left(-720\right)±1680}{2\times 24}$$

The opposite of -720 is 720.

$$x=\frac{720±1680}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{720±1680}{48}$$

Now solve the equation x=\frac{720±1680}{48} when ± is plus. Add 720 to 1680.

$$x=\frac{2400}{48}$$

Divide 2400 by 48.

$$x=50$$

Now solve the equation x=\frac{720±1680}{48} when ± is minus. Subtract 1680 from 720.

$$x=-\frac{960}{48}$$

Divide -960 by 48.

$$x=-20$$

The equation is now solved.

$$x=50 x=-20$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,30 since division by zero is not defined. Multiply both sides of the equation by x\left(x-30\right), the least common multiple of x,x-30.

$$\left(x-30\right)\times 800=x\times 800+x\left(x-30\right)\left(-24\right)$$

Use the distributive property to multiply x-30 by 800.

$$800x-24000=x\times 800+x\left(x-30\right)\left(-24\right)$$

Use the distributive property to multiply x by x-30.

$$800x-24000=x\times 800+\left(x^{2}-30x\right)\left(-24\right)$$

Use the distributive property to multiply x^{2}-30x by -24.

$$800x-24000=x\times 800-24x^{2}+720x$$

Combine x\times 800 and 720x to get 1520x.

$$800x-24000=1520x-24x^{2}$$

Subtract 1520x from both sides.

$$800x-24000-1520x=-24x^{2}$$

Combine 800x and -1520x to get -720x.

$$-720x-24000=-24x^{2}$$

Add 24x^{2} to both sides.

$$-720x-24000+24x^{2}=0$$

Add 24000 to both sides. Anything plus zero gives itself.

$$-720x+24x^{2}=24000$$

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.

$$24x^{2}-720x=24000$$

Divide both sides by 24.

$$\frac{24x^{2}-720x}{24}=\frac{24000}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\left(-\frac{720}{24}\right)x=\frac{24000}{24}$$

Divide -720 by 24.

$$x^{2}-30x=\frac{24000}{24}$$

Divide 24000 by 24.

$$x^{2}-30x=1000$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=1000+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=1000+225$$

Add 1000 to 225.

$$x^{2}-30x+225=1225$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=1225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{1225}$$

Simplify.

$$x-15=35 x-15=-35$$

Add 15 to both sides of the equation.

$$x=50 x=-20$$