Solve 2z^2=11z

$$2 z ^ { 2 } = 11 z$$

Solve for z

$$z = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

$$z=0$$

Steps Using Factoring

Subtract 11z from both sides.

$$2z^{2}-11z=0$$

Factor out z.

$$z\left(2z-11\right)=0$$

To find equation solutions, solve z=0 and 2z-11=0.

$$z=0 z=\frac{11}{2}$$

Steps Using the Quadratic Formula

Subtract 11z from both sides.

$$2z^{2}-11z=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -11 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}}}{2\times 2}$$

Take the square root of \left(-11\right)^{2}.

$$z=\frac{-\left(-11\right)±11}{2\times 2}$$

The opposite of -11 is 11.

$$z=\frac{11±11}{2\times 2}$$

Multiply 2 times 2.

$$z=\frac{11±11}{4}$$

Now solve the equation z=\frac{11±11}{4} when ± is plus. Add 11 to 11.

$$z=\frac{22}{4}$$

Reduce the fraction \frac{22}{4} to lowest terms by extracting and canceling out 2.

$$z=\frac{11}{2}$$

Now solve the equation z=\frac{11±11}{4} when ± is minus. Subtract 11 from 11.

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

Divide 0 by 4.

$$z=0$$

The equation is now solved.

$$z=\frac{11}{2} z=0$$

Steps for Completing the Square

Subtract 11z from both sides.

$$2z^{2}-11z=0$$

Divide both sides by 2.

$$\frac{2z^{2}-11z}{2}=\frac{0}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$z^{2}-\frac{11}{2}z=\frac{0}{2}$$

Divide 0 by 2.

$$z^{2}-\frac{11}{2}z=0$$

Divide -\frac{11}{2}, the coefficient of the x term, by 2 to get -\frac{11}{4}. Then add the square of -\frac{11}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-\frac{11}{2}z+\left(-\frac{11}{4}\right)^{2}=\left(-\frac{11}{4}\right)^{2}$$

Square -\frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$z^{2}-\frac{11}{2}z+\frac{121}{16}=\frac{121}{16}$$

Factor z^{2}-\frac{11}{2}z+\frac{121}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-\frac{11}{4}\right)^{2}=\frac{121}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{11}{4}\right)^{2}}=\sqrt{\frac{121}{16}}$$

Simplify.

$$z-\frac{11}{4}=\frac{11}{4} z-\frac{11}{4}=-\frac{11}{4}$$

Add \frac{11}{4} to both sides of the equation.

$$z=\frac{11}{2} z=0$$