All skills Math

Learn Index · Patterns and Algebra Grade 11–12

Radical equations

Isolate the root, square both sides, then check.

Read

Variables under a root

A radical equation has the variable inside a radical:

  • √x = 7
  • √(2x + 3) = x
  • ∛(x − 1) = 2

Steps

  1. Isolate the radical on one side.
  2. Square both sides (or cube, for cube roots).
  3. Solve the resulting equation.
  4. Check every answer in the original equation.

Example 1

√(x + 5) = 4

  • Square: x + 5 = 16
  • x = 11
  • Check: √16 = 4 ✓

Example 2: isolate first

√(3x − 2) + 4 = 9

  • Isolate: √(3x − 2) = 5
  • Square: 3x − 2 = 25
  • x = 9
  • Check: √25 + 4 = 9 ✓

Example 3: extraneous roots

√(x + 6) = x

  • Square: x + 6 = x²
  • x² − x − 6 = 0 → (x − 3)(x + 2) = 0 → x = 3 or x = −2

Check:

  • x = 3: √9 = 3 ✓
  • x = −2: √4 = 2, but the right side is −2 ✗

x = −2 is an extraneous root. Solution: x = 3.

Why does this happen? Squaring loses sign information: both 2 and −2 square to 4. The √ symbol means the principal root (non-negative), so √4 = 2, never −2.

Quick check for no solution

√(x − 1) = −3 has no solution: a principal square root can’t be negative.

Cube roots

Cube both sides; cube roots can be negative, so extraneous roots don’t arise this way.

∛(2x + 1) = −3 → 2x + 1 = −27 → x = −14

Two radicals

√(x + 7) = √(2x − 1)

  • Square: x + 7 = 2x − 1 → x = 8
  • Check: √15 = √15 ✓

Applications

Pendulum: T = 2π√(L/9.8). For T = 2 s: √(L/9.8) = 1/π → L = 9.8/π² ≈ 0.99 m.

Falling objects: t = √(2h/9.8). Solve for h given t.

Worked examples

Example 1: √x − 3 = 5 → √x = 8 → x = 64.

Example 2: 2√(x − 1) = 6 → √(x − 1) = 3 → x = 10.

Example 3: √(5 − x) = x + 1 → 5 − x = x² + 2x + 1 → x² + 3x − 4 = 0 → x = 1 or −4. Check: x = 1 ✓ (√4 = 2); x = −4: √9 = 3 but −3 ✗ → x = 1.

Common mistakes

  • Squaring before isolating: (√a + 4)² ≠ a + 16.
  • Not checking, so extraneous roots slip through.
  • Squaring only one side.

Try it yourself

  1. √(x − 2) = 6
  2. √(4x + 1) − 1 = 4
  3. √(x + 2) = x
  4. √(x + 3) = −2
  5. ∛(x + 4) = 2

Answers: 1) 38 2) 6 3) 2 (−1 is extraneous) 4) no solution 5) 4

Quick recap

  • Isolate the radical, then square both sides.
  • Always check: squaring can create extraneous roots.
  • √ means the principal root, which is never negative.