To solve the equation [tex]\( x^2 - x^3 = 12 \)[/tex], follow these steps:
1. Rewrite the equation into standard form:
[tex]\[
x^2 - x^3 - 12 = 0
\][/tex]
2. Identify the polynomial:
We have a cubic polynomial equation:
[tex]\[
-x^3 + x^2 - 12 = 0
\][/tex]
3. Look for potential solutions by attempting to factorize the polynomial if possible. However, since factorization might not be straightforward for this cubic equation, we consider solving it directly.
4. Solve the polynomial equation using an appropriate method such as numerical methods, the Rational Root Theorem, or complex number solutions. For this particular problem, solving with complex roots is also necessary.
After going through the algebraic methods or using tools to solve the polynomial, the solutions to the equation
[tex]\[ x^2 - x^3 - 12 = 0 \][/tex]
are found to be:
[tex]\[ x = -2 \][/tex]
[tex]\[ x = \frac{3}{2} - \frac{\sqrt{15}i}{2} \][/tex]
[tex]\[ x = \frac{3}{2} + \frac{\sqrt{15}i}{2} \][/tex]
These solutions consist of one real root ([tex]\( x = -2 \)[/tex]) and two complex roots [tex]\(\left( x = \frac{3}{2} - \frac{\sqrt{15}i}{2} \text{ and } x = \frac{3}{2} + \frac{\sqrt{15}i}{2} \right)\)[/tex].
Thus, the detailed solutions to the given equation are:
[tex]\[ x = -2 \][/tex]
[tex]\[ x = \frac{3}{2} - \frac{\sqrt{15}i}{2} \][/tex]
[tex]\[ x = \frac{3}{2} + \frac{\sqrt{15}i}{2} \][/tex]
These are the values of [tex]\( x \)[/tex] that satisfy the original equation [tex]\( x^2 - x^3 = 12 \)[/tex].
