Solve 200x-50x^2=450

$$200 x - 50 x ^ { 2 } = 450$$

Solve for x (complex solution)

$$x=2+\sqrt{5}i\approx 2+2.236067977i$$

$$x=-\sqrt{5}i+2\approx 2-2.236067977i$$

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.

$$-50x^{2}+200x=450$$

Subtract 450 from both sides of the equation.

$$-50x^{2}+200x-450=450-450$$

Subtracting 450 from itself leaves 0.

$$-50x^{2}+200x-450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -50 for a, 200 for b, and -450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-50\right)\left(-450\right)}}{2\left(-50\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-50\right)\left(-450\right)}}{2\left(-50\right)}$$

Multiply -4 times -50.

$$x=\frac{-200±\sqrt{40000+200\left(-450\right)}}{2\left(-50\right)}$$

Multiply 200 times -450.

$$x=\frac{-200±\sqrt{40000-90000}}{2\left(-50\right)}$$

Add 40000 to -90000.

$$x=\frac{-200±\sqrt{-50000}}{2\left(-50\right)}$$

Take the square root of -50000.

$$x=\frac{-200±100\sqrt{5}i}{2\left(-50\right)}$$

Multiply 2 times -50.

$$x=\frac{-200±100\sqrt{5}i}{-100}$$

Now solve the equation x=\frac{-200±100\sqrt{5}i}{-100} when ± is plus. Add -200 to 100i\sqrt{5}.

$$x=\frac{-200+100\sqrt{5}i}{-100}$$

Divide -200+100i\sqrt{5} by -100.

$$x=-\sqrt{5}i+2$$

Now solve the equation x=\frac{-200±100\sqrt{5}i}{-100} when ± is minus. Subtract 100i\sqrt{5} from -200.

$$x=\frac{-100\sqrt{5}i-200}{-100}$$

Divide -200-100i\sqrt{5} by -100.

$$x=2+\sqrt{5}i$$

The equation is now solved.

$$x=-\sqrt{5}i+2 x=2+\sqrt{5}i$$

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.

$$-50x^{2}+200x=450$$

Divide both sides by -50.

$$\frac{-50x^{2}+200x}{-50}=\frac{450}{-50}$$

Dividing by -50 undoes the multiplication by -50.

$$x^{2}+\frac{200}{-50}x=\frac{450}{-50}$$

Divide 200 by -50.

$$x^{2}-4x=\frac{450}{-50}$$

Divide 450 by -50.

$$x^{2}-4x=-9$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=-9+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-9+4$$

Add -9 to 4.

$$x^{2}-4x+4=-5$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=-5$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-5}$$

Simplify.

$$x-2=\sqrt{5}i x-2=-\sqrt{5}i$$

Add 2 to both sides of the equation.

$$x=2+\sqrt{5}i x=-\sqrt{5}i+2$$