SAT-39: Radical Equations and Extraneous Solutions

SAT-39: Radical Equations and Extraneous Solutions

Description: A radical equation has the variable under a square root, like √(x + 2) = 3. The trick is to undo the root by squaring — but squaring can create fake answers called extraneous solutions, so checking is required.

The method

1. Isolate the radical on one side (get √… alone).
2. Square both sides to remove the root.
3. Solve the equation that remains.
4. Check every answer in the original equation — discard any that don't work.

(Oʻzbekcha: avval ildizni bir tomonda yolgʻiz qoldiring, keyin ikkala tomonni kvadratga koʻtaring.)

Why checking matters

Squaring turns both +n and −n into the same number, so it can introduce a solution the original equation never had. A square root gives a non-negative result, so any answer that makes the two sides have opposite signs is extraneous.

Rule: never trust a radical-equation answer until you plug it back in. (Oʻzbekcha: kvadratga koʻtarish soxta yechim keltirib chiqarishi mumkin — shuning uchun har bir javobni tekshiring.)

Worked Example

Solve √(x + 2) = x.

• Radical is already isolated. Square both sides: x + 2 = x².
• Rearrange: x² − x − 2 = 0 → (x − 2)(x + 1) = 0 → x = 2 or x = −1.
• Check x = 2: √4 = 2 ✓. Check x = −1: √1 = 1, but right side is −1 → 1 ≠ −1 ✗.
• So x = −1 is extraneous. The only solution is x = 2.

(Oʻzbekcha: x = −1 soxta yechim, chunki ildiz manfiy boʻla olmaydi.)

Practice

Solve √(2x + 3) = x.

Key words — Kalit soʻzlar

• Radical equation — ildizli tenglama
• Square root — kvadrat ildiz
• Radical sign (√) — ildiz belgisi
• Isolate — yolgʻiz qoldirish (ajratish)
• Square both sides — ikkala tomonni kvadratga koʻtarish
• Extraneous solution — soxta (begona) yechim
• Check / Verify — tekshirish
• Non-negative — manfiy boʻlmagan
• Rearrange — qayta tartiblash

Summary

• Isolate the radical, then square both sides to remove it.
• Solve the resulting equation (often a quadratic).
• Always check answers; throw out extraneous ones that fail the original.