# How do you solve 32p^2-2p^6<0 using a sign chart?

May 30, 2018

The solution is $p \in \left(- \infty , - 2\right) \cup \left(2 , + \infty\right)$

#### Explanation:

The inequality is

$32 {p}^{2} - 2 {p}^{6} < 0$

Factorising

$2 {p}^{2} \left(16 - {p}^{4}\right) < 0$

$2 {p}^{2} \left(4 - {p}^{2}\right) \left(4 + {p}^{2}\right) < 0$

$2 {p}^{2} \left(2 + p\right) \left(2 - p\right) \left(4 + {p}^{2}\right) < 0$

Let's build the sign chart

$\left(4 + {p}^{2}\right) > 0$

Let $f \left(p\right) = 2 {p}^{2} \left(2 + p\right) \left(2 - p\right) \left(4 + {p}^{2}\right)$

$\textcolor{w h i t e}{a a a a}$$p$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 2$$\textcolor{w h i t e}{a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a a}$$2$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$${p}^{2}$$\textcolor{w h i t e}{a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a a}$$+$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$2 + p$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a}$$+$$\textcolor{w h i t e}{a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$2 - p$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a a a a}$$+$$\textcolor{w h i t e}{a a}$color(white)(aaaaaa)+$\textcolor{w h i t e}{a}$$0$$\textcolor{w h i t e}{a a}$$-$

$\textcolor{w h i t e}{a a a a}$$f \left(p\right)$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a}$$+$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a}$$0$$\textcolor{w h i t e}{a a}$$-$

Therefore,

$f \left(p\right) < 0$ when $p \in \left(- \infty , - 2\right) \cup \left(2 , + \infty\right)$

graph{32x^2-2x^6 [-7.554, 6.49, -3.425, 3.595]}