Solve 45000=-2x^2+200x+400

$$45000=-2 { x }^{ 2 } +200x+400$$

Solve for x (complex solution)

$$x=50+30\sqrt{22}i\approx 50+140.712472795i$$

$$x=-30\sqrt{22}i+50\approx 50-140.712472795i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-2x^{2}+200x+400=45000$$

Subtract 45000 from both sides.

$$-2x^{2}+200x+400-45000=0$$

Subtract 45000 from 400 to get -44600.

$$-2x^{2}+200x-44600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 200 for b, and -44600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-2\right)\left(-44600\right)}}{2\left(-2\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-2\right)\left(-44600\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-200±\sqrt{40000+8\left(-44600\right)}}{2\left(-2\right)}$$

Multiply 8 times -44600.

$$x=\frac{-200±\sqrt{40000-356800}}{2\left(-2\right)}$$

Add 40000 to -356800.

$$x=\frac{-200±\sqrt{-316800}}{2\left(-2\right)}$$

Take the square root of -316800.

$$x=\frac{-200±120\sqrt{22}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-200±120\sqrt{22}i}{-4}$$

Now solve the equation x=\frac{-200±120\sqrt{22}i}{-4} when ± is plus. Add -200 to 120i\sqrt{22}.

$$x=\frac{-200+120\sqrt{22}i}{-4}$$

Divide -200+120i\sqrt{22} by -4.

$$x=-30\sqrt{22}i+50$$

Now solve the equation x=\frac{-200±120\sqrt{22}i}{-4} when ± is minus. Subtract 120i\sqrt{22} from -200.

$$x=\frac{-120\sqrt{22}i-200}{-4}$$

Divide -200-120i\sqrt{22} by -4.

$$x=50+30\sqrt{22}i$$

The equation is now solved.

$$x=-30\sqrt{22}i+50 x=50+30\sqrt{22}i$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-2x^{2}+200x+400=45000$$

Subtract 400 from both sides.

$$-2x^{2}+200x=45000-400$$

Subtract 400 from 45000 to get 44600.

$$-2x^{2}+200x=44600$$

Divide both sides by -2.

$$\frac{-2x^{2}+200x}{-2}=\frac{44600}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{200}{-2}x=\frac{44600}{-2}$$

Divide 200 by -2.

$$x^{2}-100x=\frac{44600}{-2}$$

Divide 44600 by -2.

$$x^{2}-100x=-22300$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=-22300+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=-22300+2500$$

Add -22300 to 2500.

$$x^{2}-100x+2500=-19800$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=-19800$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{-19800}$$

Simplify.

$$x-50=30\sqrt{22}i x-50=-30\sqrt{22}i$$

Add 50 to both sides of the equation.

$$x=50+30\sqrt{22}i x=-30\sqrt{22}i+50$$