IDNLearn.com connects you with a community of experts ready to answer your questions. Get accurate and comprehensive answers from our network of experienced professionals.
Sagot :
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]
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]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.