Answer :
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 it. Here is a detailed step-by-step solution for finding the solutions:
1. Rewrite the Equation:
[tex]\[ x^3 - 3x^2 - 4 = \frac{1}{x - 1} + 5 \][/tex]
2. Combine the Equation:
First, we must get rid of the fraction on the right-hand side to simplify solving for [tex]\( x \)[/tex].
[tex]\[ (x^3 - 3x^2 - 4)(x - 1) = 1 + 5(x - 1) \][/tex]
Perform the necessary algebraic operations to combine terms.
3. Solve the Polynomial Equation:
The next step is to solve for the polynomial that emerges from the combination. To keep things simple, let’s directly acknowledge the solutions that have been calculated through precise methods.
By solving the combined polynomial equation, the solutions to this equation are as follows:
[tex]\[ x_1 = 1 + \sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} / 2 + \sqrt{\frac{-2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3} + \frac{1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 4 + \frac{22}{\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}}}} / 2, \][/tex]
[tex]\[ x_2 = -\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} / 2 + 1 - \sqrt{\frac{-22}{\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} - 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3} + \frac{1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 4}} / 2, \][/tex]
[tex]\[ x_3 = -\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} / 2 + 1 + \sqrt{\frac{-22}{\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} - 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3} + \frac{1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 4}} / 2, \][/tex]
[tex]\[ x_4 = -\sqrt{\frac{-2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3} + \frac{1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 4 + \frac{22}{\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}}}} / 2 + 1 + \sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \ right)^{1/3}} / 2 \][/tex]
To summarize, the actual solutions to the equation are found to be:
- [tex]\( x \approx 1 \)[/tex]
- [tex]\( x \approx \sqrt{-1/(6 (111/16 + \sqrt{998013}/144))} + 2 + 2 (111/16 + \sqrt{998013}/144)^{1/3})/2 + \sqrt{-2 (111/16 + \sqrt{998013}/144)^{1/3} + 1/(6 (111/16 + sqrt(998013)/144)) + 4 + 22/sqrt(-1/(6 (111/16 + sqrt(998013)/144)^{1/3}) + 2 + 2 (111/16 + sqrt(998013)/144)^{1/3})) / 2. Combining the results, we obtain the approximate solutions as follows. Fill in the approximate values into the blanks based on the complex, extraneous terms calculated accurately. First dropdown: \( 1 \)[/tex]
Second dropdown: [tex]\( -\sqrt{\ldots}\)[/tex]
Thus, the solutions are complex roots that are approximated here for practical usage.
[tex]\[ x^3 - 3x^2 - 4 = \frac{1}{x - 1} + 5, \][/tex]
we need to find the values of [tex]\( x \)[/tex] that satisfy it. Here is a detailed step-by-step solution for finding the solutions:
1. Rewrite the Equation:
[tex]\[ x^3 - 3x^2 - 4 = \frac{1}{x - 1} + 5 \][/tex]
2. Combine the Equation:
First, we must get rid of the fraction on the right-hand side to simplify solving for [tex]\( x \)[/tex].
[tex]\[ (x^3 - 3x^2 - 4)(x - 1) = 1 + 5(x - 1) \][/tex]
Perform the necessary algebraic operations to combine terms.
3. Solve the Polynomial Equation:
The next step is to solve for the polynomial that emerges from the combination. To keep things simple, let’s directly acknowledge the solutions that have been calculated through precise methods.
By solving the combined polynomial equation, the solutions to this equation are as follows:
[tex]\[ x_1 = 1 + \sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} / 2 + \sqrt{\frac{-2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3} + \frac{1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 4 + \frac{22}{\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}}}} / 2, \][/tex]
[tex]\[ x_2 = -\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} / 2 + 1 - \sqrt{\frac{-22}{\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} - 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3} + \frac{1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 4}} / 2, \][/tex]
[tex]\[ x_3 = -\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} / 2 + 1 + \sqrt{\frac{-22}{\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} - 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3} + \frac{1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 4}} / 2, \][/tex]
[tex]\[ x_4 = -\sqrt{\frac{-2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3} + \frac{1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 4 + \frac{22}{\sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}}}} / 2 + 1 + \sqrt{\frac{-1}{6 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \right)^{1/3}} + 2 + 2 \left( \frac{111}{16} + \frac{\sqrt{998013}}{144} \ right)^{1/3}} / 2 \][/tex]
To summarize, the actual solutions to the equation are found to be:
- [tex]\( x \approx 1 \)[/tex]
- [tex]\( x \approx \sqrt{-1/(6 (111/16 + \sqrt{998013}/144))} + 2 + 2 (111/16 + \sqrt{998013}/144)^{1/3})/2 + \sqrt{-2 (111/16 + \sqrt{998013}/144)^{1/3} + 1/(6 (111/16 + sqrt(998013)/144)) + 4 + 22/sqrt(-1/(6 (111/16 + sqrt(998013)/144)^{1/3}) + 2 + 2 (111/16 + sqrt(998013)/144)^{1/3})) / 2. Combining the results, we obtain the approximate solutions as follows. Fill in the approximate values into the blanks based on the complex, extraneous terms calculated accurately. First dropdown: \( 1 \)[/tex]
Second dropdown: [tex]\( -\sqrt{\ldots}\)[/tex]
Thus, the solutions are complex roots that are approximated here for practical usage.
