Solve 200x-x^2-3.2035=0

$$200x- { x }^{ 2 } -3.2035=0$$

Solve for x

$$x=\frac{\sqrt{99967965}}{100}+100\approx 199.983981217$$

$$x=-\frac{\sqrt{99967965}}{100}+100\approx 0.016018783$$

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.

$$-x^{2}+200x-3.2035=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 200 for b, and -3.2035 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-1\right)\left(-3.2035\right)}}{2\left(-1\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-1\right)\left(-3.2035\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-200±\sqrt{40000+4\left(-3.2035\right)}}{2\left(-1\right)}$$

Multiply 4 times -3.2035.

$$x=\frac{-200±\sqrt{40000-12.814}}{2\left(-1\right)}$$

Add 40000 to -12.814.

$$x=\frac{-200±\sqrt{39987.186}}{2\left(-1\right)}$$

Take the square root of 39987.186.

$$x=\frac{-200±\frac{\sqrt{99967965}}{50}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-200±\frac{\sqrt{99967965}}{50}}{-2}$$

Now solve the equation x=\frac{-200±\frac{\sqrt{99967965}}{50}}{-2} when ± is plus. Add -200 to \frac{\sqrt{99967965}}{50}.

$$x=\frac{\frac{\sqrt{99967965}}{50}-200}{-2}$$

Divide -200+\frac{\sqrt{99967965}}{50} by -2.

$$x=-\frac{\sqrt{99967965}}{100}+100$$

Now solve the equation x=\frac{-200±\frac{\sqrt{99967965}}{50}}{-2} when ± is minus. Subtract \frac{\sqrt{99967965}}{50} from -200.

$$x=\frac{-\frac{\sqrt{99967965}}{50}-200}{-2}$$

Divide -200-\frac{\sqrt{99967965}}{50} by -2.

$$x=\frac{\sqrt{99967965}}{100}+100$$

The equation is now solved.

$$x=-\frac{\sqrt{99967965}}{100}+100 x=\frac{\sqrt{99967965}}{100}+100$$

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.

$$-x^{2}+200x-3.2035=0$$

Add 3.2035 to both sides of the equation.

$$-x^{2}+200x-3.2035-\left(-3.2035\right)=-\left(-3.2035\right)$$

Subtracting -3.2035 from itself leaves 0.

$$-x^{2}+200x=-\left(-3.2035\right)$$

Subtract -3.2035 from 0.

$$-x^{2}+200x=3.2035$$

Divide both sides by -1.

$$\frac{-x^{2}+200x}{-1}=\frac{3.2035}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{200}{-1}x=\frac{3.2035}{-1}$$

Divide 200 by -1.

$$x^{2}-200x=\frac{3.2035}{-1}$$

Divide 3.2035 by -1.

$$x^{2}-200x=-3.2035$$

Divide -200, the coefficient of the x term, by 2 to get -100. Then add the square of -100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-200x+\left(-100\right)^{2}=-3.2035+\left(-100\right)^{2}$$

Square -100.

$$x^{2}-200x+10000=-3.2035+10000$$

Add -3.2035 to 10000.

$$x^{2}-200x+10000=9996.7965$$

Factor x^{2}-200x+10000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-100\right)^{2}=9996.7965$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-100\right)^{2}}=\sqrt{9996.7965}$$

Simplify.

$$x-100=\frac{\sqrt{99967965}}{100} x-100=-\frac{\sqrt{99967965}}{100}$$

Add 100 to both sides of the equation.

$$x=\frac{\sqrt{99967965}}{100}+100 x=-\frac{\sqrt{99967965}}{100}+100$$