For x , then the number of real roots of the equation 3 x 2 - 4 | x 2 - 1 | + x - 1 = 0 is ________. [2021]

Question

For $x \in ℝ$ , then the number of real roots of the equation $3 x^{2} - 4 | x^{2} - 1 | + x - 1 = 0$ is ________. [2021]

Smart Achievers · Accepted Answer

(4)

$3 x^{2} + x - 1 = 4 | x^{2} - 1 |$

$Case 1: If x \in [- 1, 1],$

$3 x^{2} + x - 1 = - 4 x^{2} + 4$

$\Rightarrow 7 x^{2} + x - 5 = 0$ $∵ D = 141 > 0 ∴ Equation has two roots$

$Case 2: If x \in (- \infty, - 1] \cup [1, \infty)$

$3 x^{2} + x - 1 = 4 x^{2} - 4$

$\Rightarrow x^{2} - x - 3 = 0 ∵ D = 13 > 0$

$∴$ $Equation has two roots$

$So, total 4 roots.$

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