Question

Solve the cube root equation x3−12x2+16x+48−−−−−−−−−−−−−−−−−√3=x−4
.(1 point)
Responses

x=−12
x equals negative Start Fraction 1 over 2 End Fraction

x=72
x equals Start Fraction 7 over 2 End Fraction

x=−72
x equals negative Start Fraction 7 over 2 End Fraction

x=12
x equals Start Fraction 1 over 2 End Fraction

Answer 1

We have ∛(x^3 − 12x^2 + 16x + 48) = x − 4. Cubing both sides (valid for all real numbers) gives

x^3 − 12x^2 + 16x + 48 = (x − 4)^3 = x^3 − 12x^2 + 48x − 64.

Cancel x^3 − 12x^2 and simplify:
16x + 48 = 48x − 64 → −32x + 112 = 0 → x = 112/32 = 7/2.

Check in the original equation: it holds. So x = 7/2.