Solving Square Root and Other Radical Equations - Algebra-2

1. Fundamental Concepts

Radical Equation: An equation where the variable is contained within a radicand (the expression under the radical sign). The type of radical equation is determined by its index (the small number at the top left of the radical sign; the index of a square root is implicitly 2). Common types include square root equations (index 2, e.g., ) and cube root equations (index 3, e.g., ).
Domain Restriction: For even-index radicals (indices like 2, 4, which are even numbers), the radicand must be non-negative (i.e., radicand ≥ 0). For odd-index radicals (indices like 3, 5, which are odd numbers), the radicand can be positive or negative. However, after solving, the solution must still align with real-world contexts (e.g., length or area cannot be negative).
Index: A number that indicates "the root to be taken" (e.g., index 3 means taking the cube root). It is a core feature of radical equations and determines the "exponentiation degree" required to eliminate the radical sign (e.g., an index of 3 requires cubing both sides, while an index of 2 requires squaring both sides).

"Matching Exponentiation" to Eliminate Radicals: The core strategy for solving radical equations is to convert them into polynomial equations (e.g., linear equations in one variable, quadratic equations) by performing exponentiation that matches the index:
- If the index is n, raise both sides of the equation to the n-th power (e.g., index 3 → cube both sides; index 2 → square both sides) to eliminate the radical sign.
- If the equation contains multiple radicals with the same index, first isolate one radical on one side of the equal sign. After exponentiation, handle the remaining radicals. This avoids complicated calculations.
Validation Logic for Solutions: Although it is not necessary to deliberately exclude extraneous solutions, it is essential to verify whether the solution meets the "domain restriction" and "the original equation holds true":
- First, check if the radicand satisfies the non-negativity requirement (for even-index radicals).
- Then, substitute the solution back into the original equation to calculate both sides and confirm that the equation holds, ensuring the solution is valid.
Relationship Between Index and Operations: The exponentiation operation must strictly match the index. Incorrect exponentiation (e.g., squaring a cube root equation, which fails to eliminate the radical; or cubing a square root equation, which introduces higher-degree terms and increases calculation effort) will lead to errors in solving.

Question: Solve
Solution Steps:

The radical is already isolated on the left side, and the index is 2.
To eliminate the square root, square both sides:
Simplifying gives: .
Solve the linear equation:
Add 5 to both sides:
Divide both sides by 2: .
Validation: Substitute  into the original equation:
Left side: , which equals the right side.
The radicand  (satisfies the domain of square roots).
Thus,  is a valid solution.

Question: Solve
Solution Steps:

The radical is isolated on the left side, and the index is 3 (different from the easy question’s index 2).
To eliminate the cube root, cube both sides: Simplifying gives: .
Solve the linear equation:
Subtract 7 from both sides:
Divide both sides by 4: .
Validation: Substitute into the original equation:
Left side: , which matches the right side.
Cube roots have no non-negativity restriction on the radicand, so is a valid solution.

Question: Solve
Solution Steps:

Isolate the term with the rational exponent first:
Subtract 5 from both sides: Multiply both sides by (the reciprocal of ) to eliminate the coefficient:.
Convert the rational exponent to radical form and eliminate the radical:
Recall that (since ).
To eliminate the square root and the cube, first raise both sides to the -th power (the reciprocal of ):
Simplify the left side:
Calculate the right side: (or approximately 5.241, but we keep it in exact form for precision).
Solve for x: Add 4 to both sides: .
Validation:
- Domain check: The radicand in is . Since , , satisfying the even-index radical domain.
- Substitute back into the original equation:
  Left side: , which equals the right side.
  Thus, is a valid solution.

Index Priority Judgment Method: First, mark the index of the radical when encountering an equation. Clarify the "exponentiation degree" required (e.g., mark as "index 3 → cube both sides" and as "index 4 → raise to the 4th power") to avoid mistakes in subsequent operations.
Three-Step Radical Isolation Method: For equations with a mix of radicals, constants, and polynomials (e.g., $\sqrt{3x + 12}+2=12$ ):
- Step 1: Rearrange terms to move non-radical terms (e.g., +2) to the right side of the equal sign, resulting in .
- Step 2: Perform exponentiation matching the index (here, index 2 → square both sides) to convert the equation into a polynomial equation: .
- Step 3: Solve the polynomial equation and verify the validity of the solution.
Simplification Technique for Odd-Index Radicals: There is no need to worry about negative radicands for odd-index radicals (e.g., cube roots). Exponentiation can be performed directly during solving. However, note that if sign changes occur after exponentiation (e.g., , which becomes after cubing, with the solution ), the signs must be retained completely to avoid missing negative signs.
Domain Pre-Check for Even-Index Radicals: Before solving even-index radical equations, first write the inequality for the non-negativity of the radicand (e.g., for , first write , which gives ). If the solution obtained later is less than 1.5, it can be excluded directly, reducing validation steps.