To find the solutions to the equation [tex]\( x^3 + 16x - 3x^2 = 48 \)[/tex], we start by rearranging the equation in a standard form:
[tex]\[ x^3 - 3x^2 + 16x - 48 = 0 \][/tex]
Next, we can solve this cubic equation for [tex]\( x \)[/tex]. The roots of a cubic equation can be real or complex. To determine these roots, we typically factorize the polynomial or use various methods such as the Rational Root Theorem, synthetic division, or numerical solvers.
Since full factorization or root-finding methods are sometimes complex and involve multiple steps, we will directly identify the roots by examining the solutions.
In this case, the roots of the equation [tex]\( x^3 - 3x^2 + 16x - 48 = 0 \)[/tex] are:
[tex]\[ x = 3, \quad x = -4i, \quad \text{and} \quad x = 4i \][/tex]
These roots can be verified by substituting them back into the original equation.
The first root [tex]\( x = 3 \)[/tex]:
[tex]\[ 3^3 + 16 \cdot 3 - 3 \cdot 3^2 = 27 + 48 - 27 = 48 \][/tex]
This confirms that [tex]\( x = 3 \)[/tex] is indeed a solution.
The second root [tex]\( x = -4i \)[/tex]:
When substituting [tex]\( x = -4i \)[/tex] and simplifying, the imaginary terms balance out, and the equation holds true.
The third root [tex]\( x = 4i \)[/tex]:
Similarly, substituting [tex]\( x = 4i \)[/tex] and simplifying also confirms the equation holds true in the realm of complex numbers.
Thus, the solutions to the equation [tex]\( x^3 + 16x - 3x^2 = 48 \)[/tex] are:
[tex]\[ x = 3, \quad x = -4i, \quad x = 4i \][/tex]
So, the correct answer is:
[tex]\[ 3, \ -4i, \ 4i \][/tex]
