To solve the given equation:
[tex]\[ x^3 - 3x^2 - 4 = \frac{1}{x-1} + 5 \][/tex]
we will follow these steps:
1. Rewrite the equation:
Starting from the given equation:
[tex]\[ x^3 - 3x^2 - 4 = \frac{1}{x-1} + 5 \][/tex]
2. Move all terms to one side:
To find the solutions, we need to set the equation to zero:
[tex]\[ x^3 - 3x^2 - 4 - \left(\frac{1}{x-1} + 5\right) = 0 \][/tex]
Combine like terms:
[tex]\[ x^3 - 3x^2 - 4 - \frac{1}{x-1} - 5 = 0 \][/tex]
Further simplifies to:
[tex]\[ x^3 - 3x^2 - 9 - \frac{1}{x-1} = 0 \][/tex]
3. Solve the equation:
By solving the equation, we are looking for the values of [tex]\(x\)[/tex] that satisfy it.
4. Filter out real solutions:
The solutions that satisfy the equation are approximately:
[tex]\[ x = 3.68875976307085 \][/tex]
and
[tex]\[ x = 0.906725001632668 \][/tex]
5. Conclusion:
Thus, the solutions to the equation [tex]\( x^3 - 3x^2 - 4 = \frac{1}{x-1} + 5 \)[/tex] are approximately:
[tex]\[ x \approx 3.6888 \quad \text{and} \quad x \approx 0.9067 \][/tex]
So, the correct solutions for the drop-down menu would be:
[tex]\[ x = \text{3.6888} \][/tex]
and
[tex]\[ x = \text{0.9067} \][/tex]
