To solve the equation [tex]\( x^3 - 3x^2 - 4 = \frac{1}{x-1} + 5 \)[/tex], we need to find the values of [tex]\( x \)[/tex] that satisfy this equation.
Here is the step-by-step solution:
1. Rewrite the equation into a single polynomial form:
The given equation is [tex]\( x^3 - 3x^2 - 4 = \frac{1}{x-1} + 5 \)[/tex].
2. Combine terms into a single equation:
To combine these terms into a single equation, express [tex]\( \frac{1}{x-1} + 5 \)[/tex] as a common fraction and a polynomial:
[tex]\[
\frac{1}{x-1} + 5 = \frac{1 + 5(x-1)}{x-1} = \frac{1 + 5x - 5}{x-1} = \frac{5x - 4}{x-1}
\][/tex]
3. Form a common denominator and combine terms:
Multiply both sides by [tex]\( x-1 \)[/tex] to eliminate the fraction:
[tex]\[
(x-1)\left(x^3 - 3x^2 - 4\right) = 5x - 4
\][/tex]
Expand and simplify:
[tex]\[
x^4 - x^3 - 3x^3 + 3x^2 - 4x + 4 = 5x - 4
\][/tex]
Combine like terms, bringing everything to one side:
[tex]\[
x^4 - 4x^3 + 3x^2 - 9x + 8 = 0
\][/tex]
4. Solve the polynomial equation:
Solving [tex]\( x^4 - 4x^3 + 3x^2 - 9x + 8 = 0 \)[/tex] for the roots gives us the solutions (approximately):
[tex]\[
x \approx 3.6888, \quad -0.2977 - 1.5176i, \quad -0.2977 + 1.5176i, \quad 0.9067
\][/tex]
5. Extract the real solutions:
The approximate real solutions are:
[tex]\[
x \approx 3.6888 \quad \text{and} \quad x \approx 0.9067
\][/tex]
Therefore, the correct answers to the equation [tex]\( x^3 - 3x^2 - 4 = \frac{1}{x-1} + 5 \)[/tex] are:
[tex]\[
x = 3.6888 \quad \text{and} \quad x = 0.9067.
\][/tex]
