How do you solve and graph x^3<=4x^2+3x?

Jun 25, 2017

The solution is $x \in \left(- \infty , 2 - \sqrt{7}\right] \cup \left[0 , 2 + \sqrt{7}\right]$

Explanation:

We solve this inequality with a sign chart.

${x}^{3} \le 4 {x}^{2} + 3 x$

${x}^{3} - 4 {x}^{2} - 3 x \le 0$

$x \left({x}^{2} - 4 x - 3\right) \le 0$

We need the roots of the quadratic equation

${x}^{2} - 4 x - 3 = 0$

The discrimenant is

$\Delta = {b}^{2} - 4 a c = {\left(- 4\right)}^{2} - 4 \cdot \left(1\right) \cdot \left(- 3\right) = 16 + 12 = 28$

As, $\Delta > 0$, there are 2 real roots

${x}_{2} = \frac{- b + \sqrt{\Delta}}{2} = \frac{1}{2} \left(4 + \sqrt{28}\right) = 2 + \sqrt{7} = 4.646$

${x}_{1} = \frac{- b - \sqrt{\Delta}}{2} = \frac{1}{2} \left(4 + \sqrt{28}\right) = 2 - \sqrt{7} = - 0.646$

Let $f \left(x\right) = x \left(x - {x}_{1}\right) \left(x - {x}_{2}\right)$

We can build the sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$${x}_{1}$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$${x}_{2}$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$${x}_{1}$$\textcolor{w h i t e}{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}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x$$\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}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$${x}_{2}$$\textcolor{w h i t e}{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}$$-$$\textcolor{w h i t e}{a a a a}$$+$

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

Therefore,

$f \left(x\right) \le 0$ when $x \in \left(- \infty , 2 - \sqrt{7}\right] \cup \left[0 , 2 + \sqrt{7}\right]$
graph{x^3-4x^2-3x [-12.34, 12.97, -9.01, 3.65]}