# How do you solve z^2>=4(z+3) using a sign chart?

##### 1 Answer
Jul 30, 2016

$z \le - 2$ or $z \ge 6$

#### Explanation:

${z}^{2} \ge 4 \left(z + 3\right)$ can be simplified to

${z}^{2} \ge 4 z + 12$ or

${z}^{2} - 4 z - 12 \ge 0$ or

${z}^{2} - 6 z + 2 z - 12 \ge 0$ or

$z \left(z - 6\right) + 2 \left(z - 6\right) \ge 0$ or

$\left(z + 2\right) \left(z - 6\right) \ge 0$

AS this gives critical values of $z$ as $- 2$ and $6$,

the sign chart of it is

$\textcolor{w h i t e}{\times \times \times \times \times x}$ $- 2$ color(white)(xxxxx $6$

$z + 2$$\textcolor{w h i t e}{\times \times \times}$$-$$\textcolor{w h i t e}{\times \times \times}$$+$$\textcolor{w h i t e}{\times \times \times}$$+$

$z - 6$$\textcolor{w h i t e}{\times \times \times}$$-$$\textcolor{w h i t e}{\times \times \times}$$-$$\textcolor{w h i t e}{\times \times \times}$$+$
${z}^{2} - 4 z - 12$$\textcolor{w h i t e}{x}$$+$$\textcolor{w h i t e}{\times \times \times}$$-$$\textcolor{w h i t e}{\times \times \times}$$+$

Hence solution is $z \le - 2$ or $z \ge 6$

graph{x^2-4x-12 [-10, 10, -15, 15]}