Asked by h
what are the solutions of sqrt(x^2+3x+2) = sqrt(x^2-3-1)
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GPT-5 mini
AI
Interpreting the right-hand side as sqrt(x^2 - 3 - 1) = sqrt(x^2 - 4):
Both radicands must be ≥ 0:
- x^2 + 3x + 2 = (x+1)(x+2) ≥ 0 ⇒ x ≤ −2 or x ≥ −1.
- x^2 − 4 = (x−2)(x+2) ≥ 0 ⇒ x ≤ −2 or x ≥ 2.
Their intersection (domain) is x ≤ −2 or x ≥ 2.
Square both sides (both sides nonnegative), giving
x^2 + 3x + 2 = x^2 − 4 ⇒ 3x + 2 = −4 ⇒ x = −2.
Check: at x = −2 both radicands are 0, so sqrt(0)=0 — valid.
Therefore the real solution is x = −2.
(If you meant a different expression on the right, e.g. sqrt(x^2 − 3x − 1), tell me and I’ll solve that.)
Both radicands must be ≥ 0:
- x^2 + 3x + 2 = (x+1)(x+2) ≥ 0 ⇒ x ≤ −2 or x ≥ −1.
- x^2 − 4 = (x−2)(x+2) ≥ 0 ⇒ x ≤ −2 or x ≥ 2.
Their intersection (domain) is x ≤ −2 or x ≥ 2.
Square both sides (both sides nonnegative), giving
x^2 + 3x + 2 = x^2 − 4 ⇒ 3x + 2 = −4 ⇒ x = −2.
Check: at x = −2 both radicands are 0, so sqrt(0)=0 — valid.
Therefore the real solution is x = −2.
(If you meant a different expression on the right, e.g. sqrt(x^2 − 3x − 1), tell me and I’ll solve that.)
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