Find out if cubic equation has real solution or complex ones?

Question

Without knowing any solution to that equation, is there a way to quickly tell if it has 3 real solution or 1 real and 2 conjugated ones?

Answer 1

A cubic may have two extrema or none. Three roots are possible when it has extrema and they have opposite signs.

For convenience, let us consider the depressed form,

$$x^3+px+q.$$

The maxima occur at the roots of

$$3x^2+p,$$ hence they require $p$ to be negative and $$x=\pm\sqrt{-\frac p3}.$$

Then the values at these extrema are

$$\mp\frac p3\sqrt{-\frac p3}\pm p\sqrt{-\frac p3}+q=\pm\frac23p\sqrt{-\frac p3}+q$$ and their product is

$$q^2+\frac{4p^3}{27},$$ which must be negative.

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To depress the cubic

$$ax^3+bx^2+cx+d,$$ divide by $a$ and translate the argument by $\dfrac b{3a}$.