How many positive real numbers x satisfy equation x^3-3|x|+2=0 ?

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4-2.Quadratic Equations and Inequations

normal

How many positive real numbers $x$ satisfy the equation $x^3-3|x|+2=0$ ?

$1$

$3$

$4$

$6$

(KVPY-2009)

Solution

(a)

We have,

Case $I$ $x > 0 \quad x^3-3|x|+2=0$

$\therefore \quad x^3-3 x+2=0$

$\Rightarrow \quad(x-1)(x-1)(x+2)=0$

$\Rightarrow x=1,-2$

$\text { Since, } x > 0$

$\therefore x \neq-2$

$x=1$

Case $II$ $x < 0$

$\therefore \quad x^3+3 x+2=0$

Graph of $x^3+3 x+2$

Clearly, from graph.

It has one solution lie between $(-1,0)$.

$\therefore$ Positive value of $x=1$

Hence, only one solutions.

Standard 11

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