Question

Solve $x^3 -3x^2+x-3=0$

Answer 1

The solutions are $3$, $i$, and $-i$.

Answer 2

To solve the equation $x^3 - 3x^2 + x - 3 = 0$, we use factoring by grouping: $x^2(x - 3) + 1(x - 3) = 0$ Factor out the common term $(x - 3)$: $(x - 3)(x^2 + 1) = 0$ For the product to be zero, at least one factor must be zero:

1. Set the first factor to zero: $x - 3 = 0$ $x = 3$
2. Set the second factor to zero: $x^2 + 1 = 0$ $x^2 = -1$ Taking the square root of both sides, we get: $x = \pm \sqrt{-1}$ Using the imaginary unit $i$, where $i = \sqrt{-1}$: $x = \pm i$ The roots are $3$, $i$, and $-i$.