Exercise 1666Algebra › Equations and radicalsL1Added 3 Nov 2026

A cubic equation with parameter

#cubic#parameter#roots

Statement

Find all real $m$ such that $x^3-3x+m=0$ has three distinct real roots.

Hint

Study $f(x)=x^3-3x$ and find the range of $m$ for which the horizontal line $y=-m$ crosses the graph three times.

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Critical points. $f'(x)=3x^2-3=3(x-1)(x+1)$ . Critical points $x=\pm1$ .

Values. $f(-1)=2$ , $f(1)=-2$ . The function has a local max at $-1$ (value $2$ ) and a local min at $1$ (value $-2$ ).

Condition. The equation $f(x)=-m$ has three distinct roots iff $-m\in]-2,2[$ , i.e. $m\in]-2,2[$ .

Conclusion. $m\in]-2,2[$ .