Solve x^2-250x=125000

$${ x }^{ 2 } -250x = 125000$$

Solve for x

$$x=-250$$

$$x=500$$

Steps Using Factoring

Subtract 125000 from both sides.

$$x^{2}-250x-125000=0$$

To solve the equation, factor x^{2}-250x-125000 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-250 ab=-125000$$

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 -125000.

$$1,-125000 2,-62500 4,-31250 5,-25000 8,-15625 10,-12500 20,-6250 25,-5000 40,-3125 50,-2500 100,-1250 125,-1000 200,-625 250,-500$$

Calculate the sum for each pair.

$$1-125000=-124999 2-62500=-62498 4-31250=-31246 5-25000=-24995 8-15625=-15617 10-12500=-12490 20-6250=-6230 25-5000=-4975 40-3125=-3085 50-2500=-2450 100-1250=-1150 125-1000=-875 200-625=-425 250-500=-250$$

The solution is the pair that gives sum -250.

$$a=-500 b=250$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-500\right)\left(x+250\right)$$

To find equation solutions, solve x-500=0 and x+250=0.

$$x=500 x=-250$$

Steps Using Factoring By Grouping

Subtract 125000 from both sides.

$$x^{2}-250x-125000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-125000. To find a and b, set up a system to be solved.

$$a+b=-250 ab=1\left(-125000\right)=-125000$$

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 -125000.

$$1,-125000 2,-62500 4,-31250 5,-25000 8,-15625 10,-12500 20,-6250 25,-5000 40,-3125 50,-2500 100,-1250 125,-1000 200,-625 250,-500$$

Calculate the sum for each pair.

$$1-125000=-124999 2-62500=-62498 4-31250=-31246 5-25000=-24995 8-15625=-15617 10-12500=-12490 20-6250=-6230 25-5000=-4975 40-3125=-3085 50-2500=-2450 100-1250=-1150 125-1000=-875 200-625=-425 250-500=-250$$

The solution is the pair that gives sum -250.

$$a=-500 b=250$$

Rewrite x^{2}-250x-125000 as \left(x^{2}-500x\right)+\left(250x-125000\right).

$$\left(x^{2}-500x\right)+\left(250x-125000\right)$$

Factor out x in the first and 250 in the second group.

$$x\left(x-500\right)+250\left(x-500\right)$$

Factor out common term x-500 by using distributive property.

$$\left(x-500\right)\left(x+250\right)$$

To find equation solutions, solve x-500=0 and x+250=0.

$$x=500 x=-250$$

Steps Using the Quadratic Formula

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.

$$x^{2}-250x=125000$$

Subtract 125000 from both sides of the equation.

$$x^{2}-250x-125000=125000-125000$$

Subtracting 125000 from itself leaves 0.

$$x^{2}-250x-125000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -250 for b, and -125000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-250\right)±\sqrt{\left(-250\right)^{2}-4\left(-125000\right)}}{2}$$

Square -250.

$$x=\frac{-\left(-250\right)±\sqrt{62500-4\left(-125000\right)}}{2}$$

Multiply -4 times -125000.

$$x=\frac{-\left(-250\right)±\sqrt{62500+500000}}{2}$$

Add 62500 to 500000.

$$x=\frac{-\left(-250\right)±\sqrt{562500}}{2}$$

Take the square root of 562500.

$$x=\frac{-\left(-250\right)±750}{2}$$

The opposite of -250 is 250.

$$x=\frac{250±750}{2}$$

Now solve the equation x=\frac{250±750}{2} when ± is plus. Add 250 to 750.

$$x=\frac{1000}{2}$$

Divide 1000 by 2.

$$x=500$$

Now solve the equation x=\frac{250±750}{2} when ± is minus. Subtract 750 from 250.

$$x=-\frac{500}{2}$$

Divide -500 by 2.

$$x=-250$$

The equation is now solved.

$$x=500 x=-250$$

Steps for Completing the Square

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.

$$x^{2}-250x=125000$$

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.

$$x^{2}-250x+\left(-125\right)^{2}=125000+\left(-125\right)^{2}$$

Square -125.

$$x^{2}-250x+15625=125000+15625$$

Add 125000 to 15625.

$$x^{2}-250x+15625=140625$$

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}.

$$\left(x-125\right)^{2}=140625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-125\right)^{2}}=\sqrt{140625}$$

Simplify.

$$x-125=375 x-125=-375$$

Add 125 to both sides of the equation.

$$x=500 x=-250$$