Idea 1: substitution

If a complicated expression appears more than once in an equation or an inequality, we can give it a name (a new variable), solve in the new variable, and then return to the original variable. It is the first of our unifying methods: many apparently different equations reduce, with a well-chosen substitution, to a trivial quadratic equation.

In brief — Typical cases

• $(x^2)^2 - 5x^2 + 4 = 0$: set $t = x^2$ (with $t\ge 0$), solve $t^2 - 5t + 4 = 0$, find $t = 1$ or $t = 4$, return to $x$: $x = \pm 1$ or $x = \pm 2$.
• $4^x - 3\cdot 2^x + 2 = 0$: set $t = 2^x$ (with $t>0$), solve $t^2 - 3t + 2 = 0$.
• $(\log_3 x)^2 - 3\log_3 x + 2 = 0$: set $t = \log_3 x$, solve $t^2 - 3t + 2 = 0$.
• $\sqrt{x} \cdot (\sqrt{x} - 2) = 3$: set $t = \sqrt{x}$ (with $t\ge 0$), solve $t(t-2)=3$, that is $t^2 - 2t - 3 = 0$.

Caution — Constraints on the new variable

The substituted variable inherits positivity constraints (or of another kind) from the nature of the expression it represents. Do not forget them, otherwise you introduce spurious solutions.

The scheme of the method, in four steps, is always the same: start from the complicated equation in $x$, substitute, solve in the new variable, return to $x$ discarding the values that violate the constraints.

The scheme of the substitution: from the complicated form in $x$ to the new variable $t$, and back.

Links

Topics: Unifying methods
Concepts: Existence conditions · Quadratic equation · Substitution
Skills: Reasoning by cases · Solving inequalities · Solving equations