Solve 1000/x-2-840/x=5

$$\frac{ 1000 }{ x-2 } - \frac{ 840 }{ x } = 5$$

Solve for x

$$x=-8$$

$$x=42$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

$$x\times 1000-\left(x-2\right)\times 840=5x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 840.

$$x\times 1000-\left(840x-1680\right)=5x\left(x-2\right)$$

To find the opposite of 840x-1680, find the opposite of each term.

$$x\times 1000-840x+1680=5x\left(x-2\right)$$

Combine x\times 1000 and -840x to get 160x.

$$160x+1680=5x\left(x-2\right)$$

Use the distributive property to multiply 5x by x-2.

$$160x+1680=5x^{2}-10x$$

Subtract 5x^{2} from both sides.

$$160x+1680-5x^{2}=-10x$$

Add 10x to both sides.

$$160x+1680-5x^{2}+10x=0$$

Combine 160x and 10x to get 170x.

$$170x+1680-5x^{2}=0$$

Divide both sides by 5.

$$34x+336-x^{2}=0$$

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

$$-x^{2}+34x+336=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+336. To find a and b, set up a system to be solved.

$$a+b=34 ab=-336=-336$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -336.

$$-1,336 -2,168 -3,112 -4,84 -6,56 -7,48 -8,42 -12,28 -14,24 -16,21$$

Calculate the sum for each pair.

$$-1+336=335 -2+168=166 -3+112=109 -4+84=80 -6+56=50 -7+48=41 -8+42=34 -12+28=16 -14+24=10 -16+21=5$$

The solution is the pair that gives sum 34.

$$a=42 b=-8$$

Rewrite -x^{2}+34x+336 as \left(-x^{2}+42x\right)+\left(-8x+336\right).

$$\left(-x^{2}+42x\right)+\left(-8x+336\right)$$

Factor out -x in the first and -8 in the second group.

$$-x\left(x-42\right)-8\left(x-42\right)$$

Factor out common term x-42 by using distributive property.

$$\left(x-42\right)\left(-x-8\right)$$

To find equation solutions, solve x-42=0 and -x-8=0.

$$x=42 x=-8$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

$$x\times 1000-\left(x-2\right)\times 840=5x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 840.

$$x\times 1000-\left(840x-1680\right)=5x\left(x-2\right)$$

To find the opposite of 840x-1680, find the opposite of each term.

$$x\times 1000-840x+1680=5x\left(x-2\right)$$

Combine x\times 1000 and -840x to get 160x.

$$160x+1680=5x\left(x-2\right)$$

Use the distributive property to multiply 5x by x-2.

$$160x+1680=5x^{2}-10x$$

Subtract 5x^{2} from both sides.

$$160x+1680-5x^{2}=-10x$$

Add 10x to both sides.

$$160x+1680-5x^{2}+10x=0$$

Combine 160x and 10x to get 170x.

$$170x+1680-5x^{2}=0$$

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.

$$-5x^{2}+170x+1680=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 170 for b, and 1680 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-170±\sqrt{170^{2}-4\left(-5\right)\times 1680}}{2\left(-5\right)}$$

Square 170.

$$x=\frac{-170±\sqrt{28900-4\left(-5\right)\times 1680}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-170±\sqrt{28900+20\times 1680}}{2\left(-5\right)}$$

Multiply 20 times 1680.

$$x=\frac{-170±\sqrt{28900+33600}}{2\left(-5\right)}$$

Add 28900 to 33600.

$$x=\frac{-170±\sqrt{62500}}{2\left(-5\right)}$$

Take the square root of 62500.

$$x=\frac{-170±250}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-170±250}{-10}$$

Now solve the equation x=\frac{-170±250}{-10} when ± is plus. Add -170 to 250.

$$x=\frac{80}{-10}$$

Divide 80 by -10.

$$x=-8$$

Now solve the equation x=\frac{-170±250}{-10} when ± is minus. Subtract 250 from -170.

$$x=-\frac{420}{-10}$$

Divide -420 by -10.

$$x=42$$

The equation is now solved.

$$x=-8 x=42$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

$$x\times 1000-\left(x-2\right)\times 840=5x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 840.

$$x\times 1000-\left(840x-1680\right)=5x\left(x-2\right)$$

To find the opposite of 840x-1680, find the opposite of each term.

$$x\times 1000-840x+1680=5x\left(x-2\right)$$

Combine x\times 1000 and -840x to get 160x.

$$160x+1680=5x\left(x-2\right)$$

Use the distributive property to multiply 5x by x-2.

$$160x+1680=5x^{2}-10x$$

Subtract 5x^{2} from both sides.

$$160x+1680-5x^{2}=-10x$$

Add 10x to both sides.

$$160x+1680-5x^{2}+10x=0$$

Combine 160x and 10x to get 170x.

$$170x+1680-5x^{2}=0$$

Subtract 1680 from both sides. Anything subtracted from zero gives its negation.

$$170x-5x^{2}=-1680$$

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.

$$-5x^{2}+170x=-1680$$

Divide both sides by -5.

$$\frac{-5x^{2}+170x}{-5}=-\frac{1680}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{170}{-5}x=-\frac{1680}{-5}$$

Divide 170 by -5.

$$x^{2}-34x=-\frac{1680}{-5}$$

Divide -1680 by -5.

$$x^{2}-34x=336$$

Divide -34, the coefficient of the x term, by 2 to get -17. Then add the square of -17 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-34x+\left(-17\right)^{2}=336+\left(-17\right)^{2}$$

Square -17.

$$x^{2}-34x+289=336+289$$

Add 336 to 289.

$$x^{2}-34x+289=625$$

Factor x^{2}-34x+289. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-17\right)^{2}=625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-17\right)^{2}}=\sqrt{625}$$

Simplify.

$$x-17=25 x-17=-25$$

Add 17 to both sides of the equation.

$$x=42 x=-8$$