Discussion of a parametric equation

Let us see how a discussion is carried out in practice, distinguishing all the cases that the parameters can generate.

Example

Solve $a^2x + ax - ab = 0$ as $a,b\in\mathbb{R}$ vary.

We collect the $x$: $x(a^2+a) = ab$, that is $x\cdot a(a+1) = ab$.

Discussion:

• If $a\neq 0$ and $a\neq -1$: the coefficient $a(a+1)\neq 0$, hence $x = \frac{ab}{a(a+1)} = \boxed{\dfrac{b}{a+1}} \quad\text{(unique solution)}$
• If $a = 0$: the equation becomes $0 = 0$, always true $\implies$ indeterminate (infinitely many solutions).
• If $a = -1$: the equation becomes $0 = -b$, and we must distinguish further:
  • if $b=0$: $0=0 \implies$ indeterminate;
  • if $b\neq 0$: $0 = -b \neq 0 \implies$ impossible (no solution).

The example shows the general logic: we isolate the coefficient of the unknown, set it to zero to find the “critical” values of the parameters and analyse each of them separately.

Links

Topics: First-degree equations
Concepts: Parametric discussion · Impossible equation · Indeterminate equation · Literal equation · Parameter
Skills: Reasoning by cases · Solving equations