Let's consider the given equation:
[tex]\[
x^3 - 3x^2 - 4 = \frac{1}{x-1} + 5
\][/tex]
To solve for [tex]\( x \)[/tex], we simplify and solve the equation step-by-step.
1. First, move all terms to one side of the equation to set it equal to zero:
[tex]\[
x^3 - 3x^2 - 4 - \left( \frac{1}{x-1} + 5 \right) = 0
\][/tex]
2. Combine and simplify the terms on the left side:
[tex]\[
x^3 - 3x^2 - 4 - \frac{1}{x-1} - 5 = 0
\][/tex]
[tex]\[
x^3 - 3x^2 - 9 - \frac{1}{x-1} = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex] to find the roots of the equation.
After solving, we obtain the following approximate solutions for [tex]\( x \)[/tex]:
[tex]\[
x \approx 3.68876, \quad x \approx -0.297742 \pm 1.51763i, \quad \text{and} \quad x \approx 0.906725
\][/tex]
Thus, the real solutions to the equation are approximately:
[tex]\[
x \approx 3.68876 \quad \text{and} \quad x \approx 0.906725
\][/tex]
Therefore, in the given drop-down menus, you should select:
- One drop-down to [tex]\( x \approx 3.68876 \)[/tex]
- The other drop-down to [tex]\( x \approx 0.906725 \)[/tex]
