Solve 0=x-0.025x^2

$$0=x-0.025 { x }^{ 2 }$$

Solve for x

$$x=40$$

$$x=0$$

Steps Using Factoring

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

$$x-0.025x^{2}=0$$

Factor out x.

$$x\left(1-0.025x\right)=0$$

To find equation solutions, solve x=0 and 1-\frac{x}{40}=0.

$$x=0 x=40$$

Steps Using the Quadratic Formula

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

$$x-0.025x^{2}=0$$

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.

$$-0.025x^{2}+x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.025 for a, 1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}}}{2\left(-0.025\right)}$$

Take the square root of 1^{2}.

$$x=\frac{-1±1}{2\left(-0.025\right)}$$

Multiply 2 times -0.025.

$$x=\frac{-1±1}{-0.05}$$

Now solve the equation x=\frac{-1±1}{-0.05} when ± is plus. Add -1 to 1.

$$x=\frac{0}{-0.05}$$

Divide 0 by -0.05 by multiplying 0 by the reciprocal of -0.05.

$$x=0$$

Now solve the equation x=\frac{-1±1}{-0.05} when ± is minus. Subtract 1 from -1.

$$x=-\frac{2}{-0.05}$$

Divide -2 by -0.05 by multiplying -2 by the reciprocal of -0.05.

$$x=40$$

The equation is now solved.

$$x=0 x=40$$

Steps for Completing the Square

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

$$x-0.025x^{2}=0$$

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.

$$-0.025x^{2}+x=0$$

Multiply both sides by -40.

$$\frac{-0.025x^{2}+x}{-0.025}=\frac{0}{-0.025}$$

Dividing by -0.025 undoes the multiplication by -0.025.

$$x^{2}+\frac{1}{-0.025}x=\frac{0}{-0.025}$$

Divide 1 by -0.025 by multiplying 1 by the reciprocal of -0.025.

$$x^{2}-40x=\frac{0}{-0.025}$$

Divide 0 by -0.025 by multiplying 0 by the reciprocal of -0.025.

$$x^{2}-40x=0$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-40x+\left(-20\right)^{2}=\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=400$$

Factor x^{2}-40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-20\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{400}$$

Simplify.

$$x-20=20 x-20=-20$$

Add 20 to both sides of the equation.

$$x=40 x=0$$