Solve 1500/x-1500/x+250=1/2

$$\frac { 1500 } { x } - \frac { 1500 } { x + 250 } = \frac { 1 } { 2 }$$

Solve for x

$$x=-1000$$

$$x=750$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -250,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+250\right), the least common multiple of x,x+250,2.

$$\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)$$

Use the distributive property to multiply 2x+500 by 1500.

$$3000x+750000-2x\times 1500=x\left(x+250\right)$$

Multiply 2 and 1500 to get 3000.

$$3000x+750000-3000x=x\left(x+250\right)$$

Use the distributive property to multiply x by x+250.

$$3000x+750000-3000x=x^{2}+250x$$

Subtract x^{2} from both sides.

$$3000x+750000-3000x-x^{2}=250x$$

Subtract 250x from both sides.

$$3000x+750000-3000x-x^{2}-250x=0$$

Combine 3000x and -250x to get 2750x.

$$2750x+750000-3000x-x^{2}=0$$

Combine 2750x and -3000x to get -250x.

$$-250x+750000-x^{2}=0$$

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

$$-x^{2}-250x+750000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+750000. To find a and b, set up a system to be solved.

$$a+b=-250 ab=-750000=-750000$$

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 -750000.

$$1,-750000 2,-375000 3,-250000 4,-187500 5,-150000 6,-125000 8,-93750 10,-75000 12,-62500 15,-50000 16,-46875 20,-37500 24,-31250 25,-30000 30,-25000 40,-18750 48,-15625 50,-15000 60,-12500 75,-10000 80,-9375 100,-7500 120,-6250 125,-6000 150,-5000 200,-3750 240,-3125 250,-3000 300,-2500 375,-2000 400,-1875 500,-1500 600,-1250 625,-1200 750,-1000$$

Calculate the sum for each pair.

$$1-750000=-749999 2-375000=-374998 3-250000=-249997 4-187500=-187496 5-150000=-149995 6-125000=-124994 8-93750=-93742 10-75000=-74990 12-62500=-62488 15-50000=-49985 16-46875=-46859 20-37500=-37480 24-31250=-31226 25-30000=-29975 30-25000=-24970 40-18750=-18710 48-15625=-15577 50-15000=-14950 60-12500=-12440 75-10000=-9925 80-9375=-9295 100-7500=-7400 120-6250=-6130 125-6000=-5875 150-5000=-4850 200-3750=-3550 240-3125=-2885 250-3000=-2750 300-2500=-2200 375-2000=-1625 400-1875=-1475 500-1500=-1000 600-1250=-650 625-1200=-575 750-1000=-250$$

The solution is the pair that gives sum 250.

$$a=-750 b=1000$$

Rewrite -x^{2}-250x+750000 as \left(-x^{2}-750x\right)+\left(1000x+750000\right).

$$\left(-x^{2}-750x\right)+\left(1000x+750000\right)$$

Factor out x in the first and 1000 in the second group.

$$x\left(x-750\right)+1000\left(x-750\right)$$

Factor out common term x-750 by using distributive property.

$$\left(x-750\right)\left(x+1000\right)$$

To find equation solutions, solve x-750=0 and x+1000=0.

$$x=750 x=-1000$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -250,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+250\right), the least common multiple of x,x+250,2.

$$\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)$$

Use the distributive property to multiply 2x+500 by 1500.

$$3000x+750000-2x\times 1500=x\left(x+250\right)$$

Multiply 2 and 1500 to get 3000.

$$3000x+750000-3000x=x\left(x+250\right)$$

Use the distributive property to multiply x by x+250.

$$3000x+750000-3000x=x^{2}+250x$$

Subtract x^{2} from both sides.

$$3000x+750000-3000x-x^{2}=250x$$

Subtract 250x from both sides.

$$3000x+750000-3000x-x^{2}-250x=0$$

Combine 3000x and -250x to get 2750x.

$$2750x+750000-3000x-x^{2}=0$$

Combine 2750x and -3000x to get -250x.

$$-250x+750000-x^{2}=0$$

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.

$$-x^{2}-250x+750000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -250 for b, and 750000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-250\right)±\sqrt{\left(-250\right)^{2}-4\left(-1\right)\times 750000}}{2\left(-1\right)}$$

Square -250.

$$x=\frac{-\left(-250\right)±\sqrt{62500-4\left(-1\right)\times 750000}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-250\right)±\sqrt{62500+4\times 750000}}{2\left(-1\right)}$$

Multiply 4 times 750000.

$$x=\frac{-\left(-250\right)±\sqrt{62500+3000000}}{2\left(-1\right)}$$

Add 62500 to 3000000.

$$x=\frac{-\left(-250\right)±\sqrt{3062500}}{2\left(-1\right)}$$

Take the square root of 3062500.

$$x=\frac{-\left(-250\right)±1750}{2\left(-1\right)}$$

The opposite of -250 is 250.

$$x=\frac{250±1750}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{250±1750}{-2}$$

Now solve the equation x=\frac{250±1750}{-2} when ± is plus. Add 250 to 1750.

$$x=\frac{2000}{-2}$$

Divide 2000 by -2.

$$x=-1000$$

Now solve the equation x=\frac{250±1750}{-2} when ± is minus. Subtract 1750 from 250.

$$x=-\frac{1500}{-2}$$

Divide -1500 by -2.

$$x=750$$

The equation is now solved.

$$x=-1000 x=750$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -250,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+250\right), the least common multiple of x,x+250,2.

$$\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)$$

Use the distributive property to multiply 2x+500 by 1500.

$$3000x+750000-2x\times 1500=x\left(x+250\right)$$

Multiply 2 and 1500 to get 3000.

$$3000x+750000-3000x=x\left(x+250\right)$$

Use the distributive property to multiply x by x+250.

$$3000x+750000-3000x=x^{2}+250x$$

Subtract x^{2} from both sides.

$$3000x+750000-3000x-x^{2}=250x$$

Subtract 250x from both sides.

$$3000x+750000-3000x-x^{2}-250x=0$$

Combine 3000x and -250x to get 2750x.

$$2750x+750000-3000x-x^{2}=0$$

Subtract 750000 from both sides. Anything subtracted from zero gives its negation.

$$2750x-3000x-x^{2}=-750000$$

Combine 2750x and -3000x to get -250x.

$$-250x-x^{2}=-750000$$

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.

$$-x^{2}-250x=-750000$$

Divide both sides by -1.

$$\frac{-x^{2}-250x}{-1}=-\frac{750000}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{250}{-1}\right)x=-\frac{750000}{-1}$$

Divide -250 by -1.

$$x^{2}+250x=-\frac{750000}{-1}$$

Divide -750000 by -1.

$$x^{2}+250x=750000$$

Divide 250, the coefficient of the x term, by 2 to get 125. Then add the square of 125 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+250x+125^{2}=750000+125^{2}$$

Square 125.

$$x^{2}+250x+15625=750000+15625$$

Add 750000 to 15625.

$$x^{2}+250x+15625=765625$$

Factor x^{2}+250x+15625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+125\right)^{2}=765625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+125\right)^{2}}=\sqrt{765625}$$

Simplify.

$$x+125=875 x+125=-875$$

Subtract 125 from both sides of the equation.

$$x=750 x=-1000$$