A cubic discriminates real from complex roots · source reference

Question

(i) One root of $x^3-3x+c=0$, where $c$ is real, is $2-3i$. Without using a calculator, find $c$ and the other roots. [5]

(ii) If $c$ is a nonzero purely imaginary number, explain whether $x^3-3x+c=0$ can have real roots. [1]

(iii) Instead let $c$ be real. By considering $y=x^3-3x$, find the range of $c$ for which $x^3-3x+c=0$ has only real roots. [2]

[2020/VJC/Prelim/I/5]

Source working

Printed working

(i) Since the coefficients are real, $2+3i$ is also a root. Let $a$ be the third root.

$$\begin{aligned}x^3-3x+c&=(x-(2+3i))(x-(2-3i))(x-a)\\&=((x-2)-3i)((x-2)+3i)(x-a)\\&=((x-2)^2+9)(x-a)\\&=(x^2-4x+13)(x-a).\end{aligned}$$

Comparing the coefficient of $x$: $-3=4a+13\Rightarrow a=-4$. Other roots: $-4,2+3i$. Constant terms: $c=-13a=52$.

Alternative solution:

$$(2-3i)^3-3(2-3i)+c=0,$$ $$2^3+\binom31 2^2(-3i)+\binom32 2(-3i)^2+(-3i)^3-6+9i+c=0,$$ $$8-36i-54+27i-6+9i+c=0\Rightarrow c=52.$$

The conjugate $2+3i$ is also a root. Put

$$x^3-3x+52=(x-(2+3i))(x-(2-3i))(x-a).$$

Compare constants: $52=-(2+3i)(2-3i)a=-13a$, so $a=-4$.

(ii) Not possible to have real roots. If $\lambda\in\mathbb R$ were a root, $\lambda^3-3\lambda=-c$ is impossible: LHS is real and RHS is nonzero purely imaginary.

(iii) Consider $y=x^3-3x$. For only real roots, it must intersect $y=-c$ at least twice. Hence $-2\leq-c\leq2$, equivalently $-2\leq c\leq2$. At the endpoints, a repeated real root accounts for multiplicity.

Teacher note

All-real coefficients imply complex roots occur in conjugate pairs. The margin specifically states that $(2-3i)$, conjugate of $(2+3i)$, is also a root.

Graph labels

Axes $x,y$, origin 0, extrema $(-1,2),(1,-2)$, curve $y=x^3-3x$, horizontal $y=-c$ and axis ticks $-4,-2,2,4,-2,2$ are retained in native SVG.

Printed answer key

(i) $c=52$; other roots $-4,2+3i$. (iii) $-2\leq c\leq2$.

Native diagrams

Provenance

2025 TJC H2 Math (9758) JC2 MSM: COMPLEX NUMBERS. Question pages 2; solution pages 10, 11. These are original document page numbers, retained as text metadata.