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### Roots of $3{x^2} - 4|{x^2} - 1| + x - 1 = 0$

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

Solution

Case I

$x \le - 1 \cup x \ge 1$

The equation becomes, $3x^2-4(x^2-1)+x-1=0$

$\Rightarrow -x^2+x+3=0$

$\Rightarrow x = \frac{{1 \mp \sqrt {13} }}{2}$

Both above values satisfy the condition $x \le - 1 \cup x \ge 1$ and hence are accepted.

Case II

$-1 < x < 1$

The equation becomes, $3x^2-4(1-x^2)+x-1=0$

$\Rightarrow 7x^2+x-5=0$

$x = \frac{{ - 1 \pm \sqrt {141} }}{{14}}$

Both above values satisfy the condition $-1 < x < 1$ and hence are accepted.

So, 4 solutions in total.

### Sum of the coefficients in the expansion of $(x+y)^n$ ....

If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$