Solve 5z^2=4z

$$5 z ^ { 2 } = 4 z$$

Solve for z

$$z=\frac{4}{5}=0.8$$

$$z=0$$

Steps Using Factoring

Subtract 4z from both sides.

$$5z^{2}-4z=0$$

Factor out z.

$$z\left(5z-4\right)=0$$

To find equation solutions, solve z=0 and 5z-4=0.

$$z=0 z=\frac{4}{5}$$

Steps Using the Quadratic Formula

Subtract 4z from both sides.

$$5z^{2}-4z=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -4 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}}}{2\times 5}$$

Take the square root of \left(-4\right)^{2}.

$$z=\frac{-\left(-4\right)±4}{2\times 5}$$

The opposite of -4 is 4.

$$z=\frac{4±4}{2\times 5}$$

Multiply 2 times 5.

$$z=\frac{4±4}{10}$$

Now solve the equation z=\frac{4±4}{10} when ± is plus. Add 4 to 4.

$$z=\frac{8}{10}$$

Reduce the fraction \frac{8}{10} to lowest terms by extracting and canceling out 2.

$$z=\frac{4}{5}$$

Now solve the equation z=\frac{4±4}{10} when ± is minus. Subtract 4 from 4.

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

Divide 0 by 10.

$$z=0$$

The equation is now solved.

$$z=\frac{4}{5} z=0$$

Steps for Completing the Square

Subtract 4z from both sides.

$$5z^{2}-4z=0$$

Divide both sides by 5.

$$\frac{5z^{2}-4z}{5}=\frac{0}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$z^{2}-\frac{4}{5}z=\frac{0}{5}$$

Divide 0 by 5.

$$z^{2}-\frac{4}{5}z=0$$

Divide -\frac{4}{5}, the coefficient of the x term, by 2 to get -\frac{2}{5}. Then add the square of -\frac{2}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-\frac{4}{5}z+\left(-\frac{2}{5}\right)^{2}=\left(-\frac{2}{5}\right)^{2}$$

Square -\frac{2}{5} by squaring both the numerator and the denominator of the fraction.

$$z^{2}-\frac{4}{5}z+\frac{4}{25}=\frac{4}{25}$$

Factor z^{2}-\frac{4}{5}z+\frac{4}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-\frac{2}{5}\right)^{2}=\frac{4}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{2}{5}\right)^{2}}=\sqrt{\frac{4}{25}}$$

Simplify.

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

Add \frac{2}{5} to both sides of the equation.

$$z=\frac{4}{5} z=0$$