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7-(sqrt4x+5)=x+3
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GPT-5 mini
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Interpretation (most likely): 7 − √(4x + 5) = x + 3.
1. Rearrange: √(4x+5) = 4 − x.
2. Domain: 4x+5 ≥ 0 ⇒ x ≥ −5/4, and RHS ≥ 0 ⇒ x ≤ 4.
3. Square both sides: 4x + 5 = (4 − x)^2 = x^2 − 8x + 16 ⇒ x^2 − 12x + 11 = 0.
4. Solve: x = (12 ± 10)/2 ⇒ x = 11 or x = 1.
5. Check against domain and original equation: x = 11 gives RHS 4 − 11 = −7 (not allowed) and does not satisfy the original equation; x = 1 gives 7 − √9 = 7 − 3 = 4 = 1 + 3, so it works.
Solution: x = 1.
(If the equation instead meant 7 − (√(4x) + 5) = x + 3, there is no solution.)
1. Rearrange: √(4x+5) = 4 − x.
2. Domain: 4x+5 ≥ 0 ⇒ x ≥ −5/4, and RHS ≥ 0 ⇒ x ≤ 4.
3. Square both sides: 4x + 5 = (4 − x)^2 = x^2 − 8x + 16 ⇒ x^2 − 12x + 11 = 0.
4. Solve: x = (12 ± 10)/2 ⇒ x = 11 or x = 1.
5. Check against domain and original equation: x = 11 gives RHS 4 − 11 = −7 (not allowed) and does not satisfy the original equation; x = 1 gives 7 − √9 = 7 − 3 = 4 = 1 + 3, so it works.
Solution: x = 1.
(If the equation instead meant 7 − (√(4x) + 5) = x + 3, there is no solution.)
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