Solve for x
x=-1000
x=750
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\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)
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.
3000x+750000-2x\times 1500=x\left(x+250\right)
Use the distributive property to multiply 2x+500 by 1500.
3000x+750000-3000x=x\left(x+250\right)
Multiply 2 and 1500 to get 3000.
3000x+750000-3000x=x^{2}+250x
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=0
Subtract 250x from both sides.
2750x+750000-3000x-x^{2}=0
Combine 3000x and -250x to get 2750x.
-250x+750000-x^{2}=0
Combine 2750x and -3000x to get -250x.
-x^{2}-250x+750000=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=-250 ab=-750000=-750000
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.
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
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=-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
Calculate the sum for each pair.
a=-750 b=1000
The solution is the pair that gives sum 250.
\left(-x^{2}-750x\right)+\left(1000x+750000\right)
Rewrite -x^{2}-250x+750000 as \left(-x^{2}-750x\right)+\left(1000x+750000\right).
x\left(x-750\right)+1000\left(x-750\right)
Factor out x in the first and 1000 in the second group.
\left(x-750\right)\left(x+1000\right)
Factor out common term x-750 by using distributive property.
x=750 x=-1000
To find equation solutions, solve x-750=0 and x+1000=0.
\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)
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.
3000x+750000-2x\times 1500=x\left(x+250\right)
Use the distributive property to multiply 2x+500 by 1500.
3000x+750000-3000x=x\left(x+250\right)
Multiply 2 and 1500 to get 3000.
3000x+750000-3000x=x^{2}+250x
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=0
Subtract 250x from both sides.
2750x+750000-3000x-x^{2}=0
Combine 3000x and -250x to get 2750x.
-250x+750000-x^{2}=0
Combine 2750x and -3000x to get -250x.
-x^{2}-250x+750000=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=\frac{-\left(-250\right)±\sqrt{\left(-250\right)^{2}-4\left(-1\right)\times 750000}}{2\left(-1\right)}
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{62500-4\left(-1\right)\times 750000}}{2\left(-1\right)}
Square -250.
x=\frac{-\left(-250\right)±\sqrt{62500+4\times 750000}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-\left(-250\right)±\sqrt{62500+3000000}}{2\left(-1\right)}
Multiply 4 times 750000.
x=\frac{-\left(-250\right)±\sqrt{3062500}}{2\left(-1\right)}
Add 62500 to 3000000.
x=\frac{-\left(-250\right)±1750}{2\left(-1\right)}
Take the square root of 3062500.
x=\frac{250±1750}{2\left(-1\right)}
The opposite of -250 is 250.
x=\frac{250±1750}{-2}
Multiply 2 times -1.
x=\frac{2000}{-2}
Now solve the equation x=\frac{250±1750}{-2} when ± is plus. Add 250 to 1750.
x=-1000
Divide 2000 by -2.
x=-\frac{1500}{-2}
Now solve the equation x=\frac{250±1750}{-2} when ± is minus. Subtract 1750 from 250.
x=750
Divide -1500 by -2.
x=-1000 x=750
The equation is now solved.
\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)
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.
3000x+750000-2x\times 1500=x\left(x+250\right)
Use the distributive property to multiply 2x+500 by 1500.
3000x+750000-3000x=x\left(x+250\right)
Multiply 2 and 1500 to get 3000.
3000x+750000-3000x=x^{2}+250x
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=0
Subtract 250x from both sides.
2750x+750000-3000x-x^{2}=0
Combine 3000x and -250x to get 2750x.
2750x-3000x-x^{2}=-750000
Subtract 750000 from both sides. Anything subtracted from zero gives its negation.
-250x-x^{2}=-750000
Combine 2750x and -3000x to get -250x.
-x^{2}-250x=-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.
\frac{-x^{2}-250x}{-1}=-\frac{750000}{-1}
Divide both sides by -1.
x^{2}+\left(-\frac{250}{-1}\right)x=-\frac{750000}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}+250x=-\frac{750000}{-1}
Divide -250 by -1.
x^{2}+250x=750000
Divide -750000 by -1.
x^{2}+250x+125^{2}=750000+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=750000+15625
Square 125.
x^{2}+250x+15625=765625
Add 750000 to 15625.
\left(x+125\right)^{2}=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}.
\sqrt{\left(x+125\right)^{2}}=\sqrt{765625}
Take the square root of both sides of the equation.
x+125=875 x+125=-875
Simplify.
x=750 x=-1000
Subtract 125 from both sides of the equation.
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