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$3 = \fraction{1500}{x} - \fraction{1500}{x + 250} $
Solve for x
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3x\left(x+250\right)=\left(x+250\right)\times 1500-x\times 1500
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 x\left(x+250\right), the least common multiple of x,x+250.
3x^{2}+750x=\left(x+250\right)\times 1500-x\times 1500
Use the distributive property to multiply 3x by x+250.
3x^{2}+750x=1500x+375000-x\times 1500
Use the distributive property to multiply x+250 by 1500.
3x^{2}+750x-1500x=375000-x\times 1500
Subtract 1500x from both sides.
3x^{2}-750x=375000-x\times 1500
Combine 750x and -1500x to get -750x.
3x^{2}-750x-375000=-x\times 1500
Subtract 375000 from both sides.
3x^{2}-750x-375000+x\times 1500=0
Add x\times 1500 to both sides.
3x^{2}+750x-375000=0
Combine -750x and x\times 1500 to get 750x.
x=\frac{-750±\sqrt{750^{2}-4\times 3\left(-375000\right)}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 750 for b, and -375000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-750±\sqrt{562500-4\times 3\left(-375000\right)}}{2\times 3}
Square 750.
x=\frac{-750±\sqrt{562500-12\left(-375000\right)}}{2\times 3}
Multiply -4 times 3.
x=\frac{-750±\sqrt{562500+4500000}}{2\times 3}
Multiply -12 times -375000.
x=\frac{-750±\sqrt{5062500}}{2\times 3}
Add 562500 to 4500000.
x=\frac{-750±2250}{2\times 3}
Take the square root of 5062500.
x=\frac{-750±2250}{6}
Multiply 2 times 3.
x=\frac{1500}{6}
Now solve the equation x=\frac{-750±2250}{6} when ± is plus. Add -750 to 2250.
x=250
Divide 1500 by 6.
x=-\frac{3000}{6}
Now solve the equation x=\frac{-750±2250}{6} when ± is minus. Subtract 2250 from -750.
x=-500
Divide -3000 by 6.
x=250 x=-500
The equation is now solved.
3x\left(x+250\right)=\left(x+250\right)\times 1500-x\times 1500
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 x\left(x+250\right), the least common multiple of x,x+250.
3x^{2}+750x=\left(x+250\right)\times 1500-x\times 1500
Use the distributive property to multiply 3x by x+250.
3x^{2}+750x=1500x+375000-x\times 1500
Use the distributive property to multiply x+250 by 1500.
3x^{2}+750x-1500x=375000-x\times 1500
Subtract 1500x from both sides.
3x^{2}-750x=375000-x\times 1500
Combine 750x and -1500x to get -750x.
3x^{2}-750x+x\times 1500=375000
Add x\times 1500 to both sides.
3x^{2}+750x=375000
Combine -750x and x\times 1500 to get 750x.
\frac{3x^{2}+750x}{3}=\frac{375000}{3}
Divide both sides by 3.
x^{2}+\frac{750}{3}x=\frac{375000}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+250x=\frac{375000}{3}
Divide 750 by 3.
x^{2}+250x=125000
Divide 375000 by 3.
x^{2}+250x+125^{2}=125000+125^{2}
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+15625=125000+15625
Square 125.
x^{2}+250x+15625=140625
Add 125000 to 15625.
\left(x+125\right)^{2}=140625
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}.
\sqrt{\left(x+125\right)^{2}}=\sqrt{140625}
Take the square root of both sides of the equation.
x+125=375 x+125=-375
Simplify.
x=250 x=-500
Subtract 125 from both sides of the equation.