Solving Irrational Equations
1. Definition
An irrational equation is an equation in which the variable appears under a radical sign (square root, cube root, etc.) or equivalently in the base of a power with a fractional exponent.
Examples of irrational equations:
- $\sqrt{x - 2} = 5$
- $\sqrt[3]{x^2 - 8} + \sqrt{x + 24} = 4$
- $\sqrt[4]{x^2 + 9x + 29} = 3$
⚠️ Not irrational: The equation $x^2 - 2\sqrt{5}\,x + 3 = 0$ is NOT irrational because $-2\sqrt{5}$ is just a numeric coefficient — the variable $x$ does not appear under any radical.
2. Core Method — Raising Both Sides to the n-th Power
The fundamental technique for solving irrational equations is to raise both sides to the $n$-th power to eliminate the radical of index $n$.
$$\sqrt[n]{f(x)} = (\text{something}) \quad \Longrightarrow \quad f(x) = (\text{something})^n$$
🔴 CRITICAL WARNING: Raising both sides to an even power may introduce extraneous (false) roots. Therefore, verification of every solution in the original equation is mandatory.
Why do extraneous roots appear? Because squaring is not a reversible operation: $(-3)^2 = 3^2 = 9$, so squaring "forgets" the sign.