Question

How do you find the roots of the equations x^3-12x^2-2x+24 and x^3-5x^2+4x?

Answer 1

How do you find the roots of the equations $x^{3} - 12x^{2} - 2x + 24$ and $x^{3} - 5x^{2} + 4x$.

**Solution:**

$$x^{3} - 12x^{2} - 2x + 24 = 0$$ $$x^{2}(x - 12) - 2(x - 12) = 0$$ $$(x - 12)(x^{2} - 2) = 0$$ $$(x - 12)\left(x - \sqrt{2}\right)\left(x + \sqrt{2}\right) = 0$$

So $x_{1} = -\sqrt{2}, x_{2} = \sqrt{2}, x_{3} = 12$.

$$x^{3} - 5x^{2} + 4x = 0$$ $$x(x^{2} - 5x + 4) = 0$$ $$x(x - 4)(x - 1) = 0$$

So $x_{1} = 0, x_{2} = 1, x_{3} = 4$.

**Answer:** roots of equation $x^{3} - 12x^{2} - 2x + 24 = 0$ are $x_{1} = -\sqrt{2}, x_{2} = \sqrt{2}, x_{3} = 12$, roots of equation $x^{3} - 5x^{2} + 4x = 0$ are $x_{1} = 0, x_{2} = 1, x_{3} = 4$.