Solve z^2+z+1=2z^2+z+2z+1

$${ z }^{ 2 } +z+1 = 2 { z }^{ 2 } +z+2z+1$$

Solve for z

$$z=-2$$

$$z=0$$

Steps Using Factoring

Combine z and 2z to get 3z.

$$z^{2}+z+1=2z^{2}+3z+1$$

Subtract 2z^{2} from both sides.

$$z^{2}+z+1-2z^{2}=3z+1$$

Combine z^{2} and -2z^{2} to get -z^{2}.

$$-z^{2}+z+1=3z+1$$

Subtract 3z from both sides.

$$-z^{2}+z+1-3z=1$$

Combine z and -3z to get -2z.

$$-z^{2}-2z+1=1$$

Subtract 1 from both sides.

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

Subtract 1 from 1 to get 0.

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

Factor out z.

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

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

$$z=0 z=-2$$

Steps Using the Quadratic Formula

Combine z and 2z to get 3z.

$$z^{2}+z+1=2z^{2}+3z+1$$

Subtract 2z^{2} from both sides.

$$z^{2}+z+1-2z^{2}=3z+1$$

Combine z^{2} and -2z^{2} to get -z^{2}.

$$-z^{2}+z+1=3z+1$$

Subtract 3z from both sides.

$$-z^{2}+z+1-3z=1$$

Combine z and -3z to get -2z.

$$-z^{2}-2z+1=1$$

Subtract 1 from both sides.

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

Subtract 1 from 1 to get 0.

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

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

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

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

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

The opposite of -2 is 2.

$$z=\frac{2±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$z=\frac{2±2}{-2}$$

Now solve the equation z=\frac{2±2}{-2} when ± is plus. Add 2 to 2.

$$z=\frac{4}{-2}$$

Divide 4 by -2.

$$z=-2$$

Now solve the equation z=\frac{2±2}{-2} when ± is minus. Subtract 2 from 2.

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

Divide 0 by -2.

$$z=0$$

The equation is now solved.

$$z=-2 z=0$$

Steps for Completing the Square

Combine z and 2z to get 3z.

$$z^{2}+z+1=2z^{2}+3z+1$$

Subtract 2z^{2} from both sides.

$$z^{2}+z+1-2z^{2}=3z+1$$

Combine z^{2} and -2z^{2} to get -z^{2}.

$$-z^{2}+z+1=3z+1$$

Subtract 3z from both sides.

$$-z^{2}+z+1-3z=1$$

Combine z and -3z to get -2z.

$$-z^{2}-2z+1=1$$

Subtract 1 from both sides.

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

Subtract 1 from 1 to get 0.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

$$z^{2}+\left(-\frac{2}{-1}\right)z=\frac{0}{-1}$$

Divide -2 by -1.

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

Divide 0 by -1.

$$z^{2}+2z=0$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}+2z+1^{2}=1^{2}$$

Square 1.

$$z^{2}+2z+1=1$$

Factor z^{2}+2z+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z+1\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z+1\right)^{2}}=\sqrt{1}$$

Simplify.

$$z+1=1 z+1=-1$$

Subtract 1 from both sides of the equation.

$$z=0 z=-2$$