Question

For $x \in R$, the number of real roots of the equation $3 x^{2}-4\left|x^{2}-1\right|+x-1=0$ is ______

Answer 1

$3x^2 – 4|x^2 – 1| + x – 1 = 0$

Let $x ∈ [-1, 1]$

$⇒$ $3x^2 – 4(-x^2+1) + x – 1 = 0$

$⇒$ $3x^2 + 4x^2 – 4 + x – 1 = 0$

⇒ $$7x^2 + x – 5 = 0

$⇒$ $x = -1±\frac{\sqrt{(1+140)}}{2}$

Both values are acceptable.

Let$x ∈ (-∞, -1) ⋃ (1, ∞)$

$x^2-4(x^2-1)+x-1 = 0$

$⇒$ $x^2-x-3 = 0$

$⇒$ $x = 1±\frac{\sqrt{(1+12)}}{2}$

Again both are acceptable.

Hence, total number of solution =4.