How to find the roots of this equation

Question

I am trying to find the intercepts of:

$$y = 2 + 3x^2 - x^3$$

I have that the $y$-intercept is at $y=2$ so the point $(0,2)$ is on the graph.

But I'm having trouble finding the $x$-intercept. Factored, I have this equation:

$$3x^2 - x^3 = -2$$

$$x^2(3-x) = -2$$

Where can I go from here? Is there a simple way to do this by hand?

Same goes for this equation. What are the $x$-intercepts?

$$y = x^4 - 8x^2 + 8$$

I can't use the quadratic formula here right?

Answer 1

For the second equation:$$y=x^4-8x^2+8$$ let $A=x^2$.

$$y=x^4-8x^2+8\implies y=A^2-8A+8$$

$y=A^2-8A+8$ is in quadratic form, you can solve it like a quadratic equation.

Once you find the value of $A$, substitute $A$ for $x^2$, to find the possible values of $x$.

Step By Step. Solve for $A$ in:

$$0=A^2-8A+8$$

What do you get?

$$A=4\pm\sqrt{16-8}=4\pm2\sqrt 2$$ Substituting $x^2$ for $A$, you get: $$x^2=4\pm2\sqrt 2$$ $$x=\pm\sqrt{4\pm2\sqrt 2}$$

Answer 2

For the first by hand you must refer to the depressed cubic equation formula.

For the second just take $x^2=t$ to reduce to a quadratic equation.