Skill: radical-equations

Radical & Rational Equations

Squaring both sides can plant fake answers — so every radical equation ends with a check in the original. Learn to isolate, power, and verify; then do the same for rational equations, where forbidden denominators play the same trick.

1 Understand

The core idea in plain language.

What it is. A radical equation has a variable under a root: isolate the radical, then raise both sides to the matching power. Squaring is not reversible — it can create extraneous solutions that fail the original equation, so you must check every candidate. A rational equation has variables in denominators: multiply by the LCD, solve, then reject any value that zeroes a denominator.

Why it matters. The check is the method — without it, squaring lies to you. The same discipline (solve, then verify against the original) protects every equation with restricted domains.

Where it is used. Distance/speed formulas with roots · work-rate problems · geometry with the Pythagorean theorem.

2 See It

Diagrams that make the idea visual.

Squaring can lie: the extraneous root

√(x + 5) = x − 1: squaring gives x + 5 = x² − 2x + 1, so x = 4 or x = −1. But x = −1 fails the original: √4 = 2, not −2. The check catches the impostor.

√(x + 5) = x − 1: x = 4 checks, x = −1 is extraneous.

Rational exponents are radicals

x^2/3 means the cube root of x²: the denominator is the root, the numerator is the power. Reading exponents this way turns (x^1/2)² = x into obvious algebra.

x^(2/3) = cube root of x²: denominator = root, numerator = power.

3 Worked Examples

Follow each step. The pattern is always the same.

Example 1 Solve: √(2x + 3) = 5

The radical is isolated — square both sides: 2x + 3 = 25.
2x = 22 → x = 11.
Check: √(22 + 3) = √25 = 5 (checks).

x = 11

Check: Substitution into the original confirms it.

Example 2 Isolate first: √x + 4 = 10

The radical is NOT isolated — subtract 4 first: √x = 6.
Now square: x = 36.
Check: √36 + 4 = 6 + 4 = 10 (checks).

x = 36

Check: Substitution into the original confirms it.

Example 3 Solve: ∛(3x − 2) = 2

Cube both sides: 3x − 2 = 8.
3x = 10 → x = 10/3.

x = 10/3

Check: ∛(10 − 2) = ∛8 = 2 (checks). Odd roots never create extraneous solutions.

Example 4 No solution: √x = −4

A (principal) square root is never negative.
√x = −4 has no solution — do not even square.

No solution

Check: √x ≥ 0 for all real x, so √x = −4 is impossible.

4 Common Mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

Mistake 1: skipping the check

Wrong

Solving √(x + 5) = x − 1 and reporting x = 4, −1.

Right

x = 4 only — x = −1 fails the original equation.

Rule: every candidate gets checked in the ORIGINAL equation.

Mistake 2: squaring before isolating

Wrong

Squaring √(x) + 3 = 7 term-by-term: x + 9 = 49.

Right

Isolate first: √x = 4, then square: x = 16.

Rule: the radical must stand alone before you power both sides.

Mistake 3: keeping forbidden values

Wrong

Solving x/(x − 3) = 3/(x − 3) and answering x = 3.

Right

No solution — x = 3 zeroes the denominator.

Rule: note the forbidden values BEFORE multiplying by the LCD.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Solve √(3x − 6) = 4.

Answer

x = 22/3. Check: √(22 − 6) = √16 = 4.

2. Solve √x = −2.

Answer

No solution — a square root cannot be negative.

3. Evaluate x² − 3x + 2 at x = 3 (the checking step).

Answer

9 − 9 + 2 = 2.

Key Points to Remember

Isolate the radical BEFORE raising both sides to a power.
Squaring can create extraneous solutions — check every candidate in the original.
A square root equals a negative number → no solution, immediately.
Odd roots (cube root): no extraneous solutions, but still check arithmetic.
x^(m/n): denominator n = the root, numerator m = the power.
Rational equations: note forbidden denominators first, multiply by the LCD, reject bad values.
The check is part of the method, not optional.

Practice problems Try the tool See it on FiboMap