Question
Solve the cube root equation x3−7x2+12x+8−−−−−−−−−−−−−−−√3=x−2
. (1 point)
The solutions to the cube root equation are x±
. (1 point)
The solutions to the cube root equation are x±
Answers
GPT-5 mini
Cube both sides:
(x^3 − 7x^2 + 12x + 8) = (x − 2)^3.
Expand and simplify:
x^3 − 7x^2 + 12x + 8 = x^3 − 6x^2 + 12x − 8
⇒ −x^2 + 16 = 0 ⇒ x^2 = 16 ⇒ x = ±4.
Check: x=4 gives ∛8 = 2; x−2 = 2. x=−4 gives ∛(−216) = −6; x−2 = −6. Both work.
Answer: x = ±4.
(x^3 − 7x^2 + 12x + 8) = (x − 2)^3.
Expand and simplify:
x^3 − 7x^2 + 12x + 8 = x^3 − 6x^2 + 12x − 8
⇒ −x^2 + 16 = 0 ⇒ x^2 = 16 ⇒ x = ±4.
Check: x=4 gives ∛8 = 2; x−2 = 2. x=−4 gives ∛(−216) = −6; x−2 = −6. Both work.
Answer: x = ±4.