Condition for a cubic equation to have a single root

Question

If a cubic equation

$$ f(x) = ax^3+bx^2+cx+d$$

Is given, what is the condition for the equation to only have a single root (counting multiple roots as one)

Answer 1

There is not a unique condition: as a cubic function always has at least one real root,

• either $f'(x)\le 0$,
• or $f(x)$ has a triple root, i.e. $f(x)=a(x-\alpha)^3$,
• or $f'(x)>0$, so the function has a maximum and a minimum, and both extrema have the same sign,

Answer 2

$$f(x)=ax^3+bx^2+cx+d \implies f'(x)=3ax^2+2bx+c >0 \forall x \in R, ~if~ b^2 < 3 ac.$$ A monotonically increasing pr decreasing function has at most one root. Also $f(-\infty) f(\infty) <0$ $f(x)=0$ has at least one real root by IVT. So when $b^2<3ac$, the given cubic has only one real root.

Answer 3

You may look at its discriminant, which is zero if and only if the cubic equation has a multiple root.

For the equation $ax^3+ bx^2 + cx + d = 0$, the discriminant is $$ 18abcd – 4b^3d + b^2c^2 – 4ac^3 – 27a^2d^2. $$ Therefore the condition you seek is $$ 18abcd – 4b^3d + b^2c^2 – 4ac^3 – 27a^2d^2=0. $$

Answer 4

Seeing that there are answers involving the derivative that miss the point: It is about the Discriminant $ \Delta $. When $ \Delta \le 0 $ we have one real root.
See https://en.wikipedia.org/wiki/Cubic_equation#Nature_of_the_roots

Answer 5

Referring to this image of a specific cubic

The geometric parameters labeled on the diagram, in terms of the coefficients, are

$$\begin{align*} x_N &= \dfrac{-b}{3a} \quad \text{(abscissa of inflection point)}\\ \\ \delta^2 &= \dfrac{b^2-3ac}{9a^2} = x_N^2 - \dfrac{c}{3a} \quad \text{(x distance squared from inflection point to turning point)}\\ \\ y_N &= f(x_N) = \dfrac{2b^3}{27a^2} - \dfrac{bc}{3a} + d \quad \text{(ordinate of inflection point)}\\ \\ h &= 2a\delta^3 \quad \text{(y distance from inflection point to turning point)} \\ \end{align*}$$

The cubic can intersect the $x$-axis only once, in the following circumstances:

$$\begin{align*} h &= 0\\ \\ h &\in i\mathbb{R}\setminus 0\\ \\ \mathrm{or}\;\left|\dfrac{-y_N}{h}\right| &> 1 \\ \end{align*}$$

For the first two conditions, a $0$ or imaginary height, $h$, means the cubic won't have the two turning points, so it could never cross the $x$-axis more than once.

For the third condition, the ordinate of the inflection point of the cubic, $y_N$, is farther from the $x$-axis than the height, $h$, so the cubic could never cross the $x$-axis more than once.

Answer 6

Consider $f(x)=ax^3+bx^2+cx+d,$ with $a>0$ WLOG. Then there is a unique solution provided that either $$f'(x)=3ax^2+2bx+c$$ has almost positive sign -- that is, when $b^2-3ac\le 0$ -- or if $b^2-3ac>0,$ then we must have that $$f(r_1)f(r_2)\ge 0,$$ where $r_1,r_2$ are the roots of the quadratic equation $f'(x)=0.$