Solve 2z+4=(2z+2)(z+2)

$$2 z + 4 = ( 2 z + 2 ) ( z + 2 )$$

Solve for z

$$z=-2$$

$$z=0$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2z+2 by z+2 and combine like terms.

$$2z+4=2z^{2}+6z+4$$

Subtract 2z^{2} from both sides.

$$2z+4-2z^{2}=6z+4$$

Subtract 6z from both sides.

$$2z+4-2z^{2}-6z=4$$

Combine 2z and -6z to get -4z.

$$-4z+4-2z^{2}=4$$

Subtract 4 from both sides.

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

Subtract 4 from 4 to get 0.

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

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.

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

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

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

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

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

The opposite of -4 is 4.

$$z=\frac{4±4}{2\left(-2\right)}$$

Multiply 2 times -2.

$$z=\frac{4±4}{-4}$$

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

$$z=\frac{8}{-4}$$

Divide 8 by -4.

$$z=-2$$

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

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

Divide 0 by -4.

$$z=0$$

The equation is now solved.

$$z=-2 z=0$$

Steps for Completing the Square

Use the distributive property to multiply 2z+2 by z+2 and combine like terms.

$$2z+4=2z^{2}+6z+4$$

Subtract 2z^{2} from both sides.

$$2z+4-2z^{2}=6z+4$$

Subtract 6z from both sides.

$$2z+4-2z^{2}-6z=4$$

Combine 2z and -6z to get -4z.

$$-4z+4-2z^{2}=4$$

Subtract 4 from both sides.

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

Subtract 4 from 4 to get 0.

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

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.

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

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

$$z^{2}+\left(-\frac{4}{-2}\right)z=\frac{0}{-2}$$

Divide -4 by -2.

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

Divide 0 by -2.

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

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.

$$z=0 z=-2$$