Solve -x^2+250x-750000=0

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

Solve for x (complex solution)

$$x=-125\sqrt{47}i+125\approx 125-856.95682505i$$

$$x=125+125\sqrt{47}i\approx 125+856.95682505i$$

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.

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

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}.

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

Square 250.

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

Multiply -4 times -1.

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

Multiply 4 times -750000.

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

Add 62500 to -3000000.

$$x=\frac{-250±\sqrt{-2937500}}{2\left(-1\right)}$$

Take the square root of -2937500.

$$x=\frac{-250±250\sqrt{47}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-250±250\sqrt{47}i}{-2}$$

Now solve the equation x=\frac{-250±250\sqrt{47}i}{-2} when ± is plus. Add -250 to 250i\sqrt{47}.

$$x=\frac{-250+250\sqrt{47}i}{-2}$$

Divide -250+250i\sqrt{47} by -2.

$$x=-125\sqrt{47}i+125$$

Now solve the equation x=\frac{-250±250\sqrt{47}i}{-2} when ± is minus. Subtract 250i\sqrt{47} from -250.

$$x=\frac{-250\sqrt{47}i-250}{-2}$$

Divide -250-250i\sqrt{47} by -2.

$$x=125+125\sqrt{47}i$$

The equation is now solved.

$$x=-125\sqrt{47}i+125 x=125+125\sqrt{47}i$$

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.

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

Add 750000 to both sides of the equation.

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

Subtracting -750000 from itself leaves 0.

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

Subtract -750000 from 0.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

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.

$$x^{2}-250x+\left(-125\right)^{2}=-750000+\left(-125\right)^{2}$$

Square -125.

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

Add -750000 to 15625.

$$x^{2}-250x+15625=-734375$$

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}.

$$\left(x-125\right)^{2}=-734375$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-125\right)^{2}}=\sqrt{-734375}$$

Simplify.

$$x-125=125\sqrt{47}i x-125=-125\sqrt{47}i$$

Add 125 to both sides of the equation.

$$x=125+125\sqrt{47}i x=-125\sqrt{47}i+125$$