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Solve the radical equation √9−2x+4=x+1 Which of the following are the correct solutions?(1 point)x=2 x=3 x=2 or x=3 No real solution
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GPT-5 mini
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Interpret the equation as sqrt(9 − 2x + 4) = x + 1, i.e. sqrt(13 − 2x) = x + 1.
Domain: 13 − 2x ≥ 0 ⇒ x ≤ 6.5, and x + 1 ≥ 0 ⇒ x ≥ −1, so −1 ≤ x ≤ 6.5.
Square both sides: 13 − 2x = (x + 1)^2 = x^2 + 2x + 1
⇒ x^2 + 4x − 12 = 0 ⇒ (x + 6)(x − 2) = 0 ⇒ x = −6 or x = 2.
Check against domain: x = −6 is invalid; x = 2 is valid (gives √9 = 3 = 2+1).
So the correct solution is x = 2.
Domain: 13 − 2x ≥ 0 ⇒ x ≤ 6.5, and x + 1 ≥ 0 ⇒ x ≥ −1, so −1 ≤ x ≤ 6.5.
Square both sides: 13 − 2x = (x + 1)^2 = x^2 + 2x + 1
⇒ x^2 + 4x − 12 = 0 ⇒ (x + 6)(x − 2) = 0 ⇒ x = −6 or x = 2.
Check against domain: x = −6 is invalid; x = 2 is valid (gives √9 = 3 = 2+1).
So the correct solution is x = 2.
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