Solve 200=3x-0.01x^2

$$200 = 3 x - 0.01 x ^ { 2 }$$

Solve for x

$$x=100$$

$$x=200$$

Steps Using the Quadratic Formula

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

$$3x-0.01x^{2}=200$$

Subtract 200 from both sides.

$$3x-0.01x^{2}-200=0$$

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.

$$-0.01x^{2}+3x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.01 for a, 3 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-0.01\right)\left(-200\right)}}{2\left(-0.01\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-0.01\right)\left(-200\right)}}{2\left(-0.01\right)}$$

Multiply -4 times -0.01.

$$x=\frac{-3±\sqrt{9+0.04\left(-200\right)}}{2\left(-0.01\right)}$$

Multiply 0.04 times -200.

$$x=\frac{-3±\sqrt{9-8}}{2\left(-0.01\right)}$$

Add 9 to -8.

$$x=\frac{-3±\sqrt{1}}{2\left(-0.01\right)}$$

Take the square root of 1.

$$x=\frac{-3±1}{2\left(-0.01\right)}$$

Multiply 2 times -0.01.

$$x=\frac{-3±1}{-0.02}$$

Now solve the equation x=\frac{-3±1}{-0.02} when ± is plus. Add -3 to 1.

$$x=-\frac{2}{-0.02}$$

Divide -2 by -0.02 by multiplying -2 by the reciprocal of -0.02.

$$x=100$$

Now solve the equation x=\frac{-3±1}{-0.02} when ± is minus. Subtract 1 from -3.

$$x=-\frac{4}{-0.02}$$

Divide -4 by -0.02 by multiplying -4 by the reciprocal of -0.02.

$$x=200$$

The equation is now solved.

$$x=100 x=200$$

Steps for Completing the Square

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

$$3x-0.01x^{2}=200$$

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.

$$-0.01x^{2}+3x=200$$

Multiply both sides by -100.

$$\frac{-0.01x^{2}+3x}{-0.01}=\frac{200}{-0.01}$$

Dividing by -0.01 undoes the multiplication by -0.01.

$$x^{2}+\frac{3}{-0.01}x=\frac{200}{-0.01}$$

Divide 3 by -0.01 by multiplying 3 by the reciprocal of -0.01.

$$x^{2}-300x=\frac{200}{-0.01}$$

Divide 200 by -0.01 by multiplying 200 by the reciprocal of -0.01.

$$x^{2}-300x=-20000$$

Divide -300, the coefficient of the x term, by 2 to get -150. Then add the square of -150 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-300x+\left(-150\right)^{2}=-20000+\left(-150\right)^{2}$$

Square -150.

$$x^{2}-300x+22500=-20000+22500$$

Add -20000 to 22500.

$$x^{2}-300x+22500=2500$$

Factor x^{2}-300x+22500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-150\right)^{2}=2500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-150\right)^{2}}=\sqrt{2500}$$

Simplify.

$$x-150=50 x-150=-50$$

Add 150 to both sides of the equation.

$$x=200 x=100$$