Solve (0,1-z)z=0

$$( 0,1 - z ) z = 0$$

Solve for z

$$z=0$$

$$z=0,1$$

Steps Using Factoring

Use the distributive property to multiply 0,1-z by z.

$$0,1z-z^{2}=0$$

Factor out z.

$$z\left(0,1-z\right)=0$$

To find equation solutions, solve z=0 and 0,1-z=0.

$$z=0 z=\frac{1}{10}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 0,1-z by z.

$$0,1z-z^{2}=0$$

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.

$$-z^{2}+\frac{1}{10}z=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{1}{10} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\frac{1}{10}±\sqrt{\left(\frac{1}{10}\right)^{2}}}{2\left(-1\right)}$$

Take the square root of \left(\frac{1}{10}\right)^{2}.

$$z=\frac{-\frac{1}{10}±\frac{1}{10}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$z=\frac{-\frac{1}{10}±\frac{1}{10}}{-2}$$

Now solve the equation z=\frac{-\frac{1}{10}±\frac{1}{10}}{-2} when ± is plus. Add -\frac{1}{10} to \frac{1}{10} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$z=\frac{0}{-2}$$

Divide 0 by -2.

$$z=0$$

Now solve the equation z=\frac{-\frac{1}{10}±\frac{1}{10}}{-2} when ± is minus. Subtract \frac{1}{10} from -\frac{1}{10} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$z=-\frac{\frac{1}{5}}{-2}$$

Divide -\frac{1}{5} by -2.

$$z=\frac{1}{10}$$

The equation is now solved.

$$z=0 z=\frac{1}{10}$$

Steps for Completing the Square

Use the distributive property to multiply 0,1-z by z.

$$0,1z-z^{2}=0$$

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.

$$-z^{2}+\frac{1}{10}z=0$$

Divide both sides by -1.

$$\frac{-z^{2}+\frac{1}{10}z}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$z^{2}+\frac{\frac{1}{10}}{-1}z=\frac{0}{-1}$$

Divide \frac{1}{10} by -1.

$$z^{2}-\frac{1}{10}z=\frac{0}{-1}$$

Divide 0 by -1.

$$z^{2}-\frac{1}{10}z=0$$

Divide -\frac{1}{10}, the coefficient of the x term, by 2 to get -\frac{1}{20}. Then add the square of -\frac{1}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-\frac{1}{10}z+\left(-\frac{1}{20}\right)^{2}=\left(-\frac{1}{20}\right)^{2}$$

Square -\frac{1}{20} by squaring both the numerator and the denominator of the fraction.

$$z^{2}-\frac{1}{10}z+\frac{1}{400}=\frac{1}{400}$$

Factor z^{2}-\frac{1}{10}z+\frac{1}{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}.

$$\left(z-\frac{1}{20}\right)^{2}=\frac{1}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{1}{20}\right)^{2}}=\sqrt{\frac{1}{400}}$$

Simplify.

$$z-\frac{1}{20}=\frac{1}{20} z-\frac{1}{20}=-\frac{1}{20}$$

Add \frac{1}{20} to both sides of the equation.

$$z=\frac{1}{10} z=0$$