The number of distinct real roots of $x^{4} - 4 x^{3} + 12 x^{2} + x - 1 = 0$ is

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Solution

The given equation is
$x^{4} - 4 x^{3} + 12 x^{2} + x - 1 = 0$
$\Rightarrow x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1 + 6 x^{2} + 5 x - 2 = 0$
$\Rightarrow (x - 1)^{4} = - 6 x^{2} - 5 x + 2$
In order to find the number of solutions of the given equations, it is sufficient to find the number of point of intersections of the given curve $y = (x - 1)^{4} and y = - 6 x^{2} - 5 x + 2$ .

Clearly, there are two solutions of the given equation.

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