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

Correct Answer: 4

\(3x^2 – 4|x^2 – 1| + x – 1 = 0\)

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.