Animated Solution for Mathematics - Quadratic Equations: For $x\in \mathbb{R}$, then number of real roots of the equation $3x^2-4|x^2-1|+x-1=0$ is ____.

The equation provided is $3x^2-4|x^2-1|+x-1=0$. To simplify our perspective, we rearrange the equation to isolate the modulus term:

The behavior of the modulus function is determined by the expression inside the bars, $x^2-1$. Setting this to zero, we find the critical points at $x=1$ and $x=-1$.

These points partition the real number line into three distinct regions: the inner region $x\in [-1,1]$ and the outer regions $x\in (-\infty ,-1)\cup (1,\infty )$.

The Inner Sanctuary

In the inner region where $x\in [-1,1]$, we have $x^2-1\le 0$. Consequently, the modulus acts as a negation: $|x^2-1|=1-x^2$.

Substituting this into our equation, we obtain:

Expanding and rearranging the terms leads to the quadratic equation:

Using the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, we calculate the roots:

Since $\sqrt{141}\approx 11.87$, the roots are approximately $0.77$ and $-0.91$. Both values fall within the interval $[-1,1]$, confirming they are valid real roots.

The Outer Frontier

In the outer regions where $x\in (-\infty ,-1]\cup [1,\infty )$, we have $x^2-1\ge 0$. Here, the modulus is transparent: $|x^2-1|=x^2-1$.

Substituting this into the original equation yields:

Expanding and rearranging the terms to the right side results in:

Applying the quadratic formula again, we find:

With $\sqrt{13}\approx 3.6$, the roots are approximately $2.3$ and $-1.3$. Both values lie outside the interval $(-1,1)$, confirming they are valid real roots.

We have systematically navigated both the inner sanctuary and the outer frontier. From the first case, we identified two valid roots, and from the second case, we identified another two.

By combining these results, we conclude that the equation $3x^2-4|x^2-1|+x-1=0$ has exactly 4 real roots.