# Cubic equation (polynomial)

A cubic polynomial with real coefficients, $a x^3 + b x^2 + c x + d$, has either three real roots, or one real root and a pair of complex conjugate ones. If the latter happens, what is the explicit formula for this real solution, and what conditions can be placed on $a,b,c$ and $d$ to guarantee that the real root is positive?

• You are missing the assumption that it has real coefficients
– user208649
Jan 17, 2016 at 18:21
• Jan 17, 2016 at 18:29

For explicit formulas of the roots involving the coefficients, you may wish to consider Cardano's formula which will give it for degree 3 and 4.

• Thanks. A question: Wikipedia says if its discriminant is positive it has three distinct real roots, but in your link this is the case if the discriminant is negative?
– user305855
Jan 17, 2016 at 18:28
• Most likely means their definitions of discriminant are off by a sign. Jan 17, 2016 at 18:38

First, you need to eliminate the second degree term. For this, make $x=y+k$ with $k\in\mathbb{R}$, replace in the equation and determine $k$ in order to eliminate the second degree term.

Done that, we have an equation of the form $$ax^{3}+bx+c=0.$$ Writing $x=u+v$ and replace in the equation, you will find a quadratic equation whose roots are given in terms of $u$ and $v$ (in particular, of $x$).

Done that, it is easy to find the ultimate root.

• Very nice. Thanks.
– user305855
Jan 17, 2016 at 18:37
• I am curious to know in which conditions we have two reals roots of opposite signs? Jan 28, 2016 at 18:45
• @Zbigniew I do not know. Knows reciprocal polynomials? I'm not sure, but I think about then we are able to have two real roots of opposite signs. Jan 29, 2016 at 0:16

Case 1

If the derivative discriminant $b^2 - 3 a c > 0$

Let the roots of the derivative quadratic

$$3 a x^2 + 2 b x + c = 0$$

be

$$x_1,x_2$$

and corresponding max/ min values be

$$y_1, y_2$$

If these above $y$ values have the same sign ( both positive or both negative or $y_1 y_2 >0$) then there is a single real root and two complex conjugates.

Draw a graph of these cubic and derivative functions to ascertain why it should be so.

If $y_1, y_2$ are of opposite sign or $y_1 y_2 < 0$ then there are three real roots,and,

if one of them is zero, two real roots among them are equal.

EDIT 1:

In terms of $(a,b,c,d ),$

$\Delta > 0$ for two imaginary complex roots & one real root, and, $\Delta < 0$ for three real roots, where
$$27 \Delta=$$
$$(a c^2 ((-6 + (7 - 2 a) a) b^2 + c (-3 + a^2)^2 ) - 2 c ((-6 + 4 a) b^2 + 9 c (1 + a - a^2) ) d + 27 d^2).$$
If the derivative discriminant $b^2 - 3 a c < 0$ then we have an inflection point on the x-axis.