LEMMA

Radicals

Radical equations

ALEKS placement

Isolate the radical, square both sides, then check for extraneous roots.

What this covers

  • Get one radical alone on one side.
  • Square both entire sides — not term by term.
  • Solve the resulting equation.
  • Put every possible answer back into the original equation, because squaring can create an answer that was not there before.
  • An answer that fails this check is called extraneous. It works after squaring but not in the original equation, so leave it out.

Worked example

Worked example

Solve √(2x + 3) = x.

  1. x≥0A principal square root is never negative, so the side it equals must be nonnegative too.
  2. 2x+3=x2Square both sides.
  3. (x−3)(x+1)=0Rearrange to x² - 2x - 3 = 0 and factor.
  4. x=3; √2(−1)+3​=1=−1Check both candidates in the original. 3 works: √9 = 3. But x = -1 gives √1 = 1, not -1, so -1 is extraneous.
Another worked example

Solve √(x + 7) − 4 = 2.

  1. √x+7​=6Add 4 to both sides so the radical is alone. Squaring now, with the −4 still attached, would create a middle term.
  2. x+7=36Square both sides: √(x + 7)² = x + 7 and 6² = 36.
  3. x=29Subtract 7. Check in the original: √36 − 4 = 6 − 4 = 2.

A common mistake

A mistake Lemma catches

From √x+7​−4=2

√x+7​=−2√x+7​=6

“To clear -4 from the left side, add 4 to both sides: 2 + 4 = 6. Instead 4 was subtracted from the right side: 2 - 4 = -2, so the two sides no longer balance.”

Try one

Sample problem

Solve and check for extraneous solutions.

√x+8​=3

Practice radical equations free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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