Select the correct answer from each drop-down menu.
Consider the equation below.

[tex]x^3-3 x^2-4=\frac{1}{x-1}+5[/tex]

The solutions to the equation are approximately [tex]x=[/tex] [...dropdown...] and [tex]x=[/tex] [...dropdown...]

Rewrite the equation as a polynomial: x 4 − 4 x 3 + 3 x 2 − 9 x + 8 = 0 .
Find the approximate roots using numerical methods: x ≈ 0.9067 and x ≈ 3.6888 .
Compare the roots with the given options: 0.91, -5.72, and 0.80.
Select the closest option to one of the roots: 0.91 ​ .

Explanation

Problem Analysis We are given the equation x 3 − 3 x 2 − 4 = x − 1 1 ​ + 5 and asked to find the approximate solutions for x .

Rewriting the Equation First, let's rewrite the equation as x 3 − 3 x 2 − 4 − x − 1 1 ​ − 5 = 0 , which simplifies to x 3 − 3 x 2 − 9 − x − 1 1 ​ = 0 .

Multiplying by (x-1) Multiply both sides by ( x − 1 ) to get ( x − 1 ) ( x 3 − 3 x 2 − 9 ) − 1 = 0 .

Expanding the Equation Expand the equation to get x 4 − 3 x 3 − 9 x − x 3 + 3 x 2 + 9 − 1 = 0 , which simplifies to x 4 − 4 x 3 + 3 x 2 − 9 x + 8 = 0 .

Defining f(x) Let f ( x ) = x 4 − 4 x 3 + 3 x 2 − 9 x + 8 . We need to find the roots of f ( x ) = 0 .

Finding Approximate Roots Using a numerical method or a calculator, we find the approximate roots of the equation f ( x ) = 0 to be approximately x = 0.9067 and x = 3.6888 .

Checking the Options From the given options, the closest value to 0.9067 is 0.91. However, there is no option close to 3.6888. Let's verify the root x = 0.91 in the original equation.

Verifying Roots The solutions to the equation are approximately x = 0.91 and x = 3.69 . Since 3.69 is not an option, we will look for another root. We can use the calculate_function_approximate_real_valued_roots tool to find the roots of the equation x 4 − 4 x 3 + 3 x 2 − 9 x + 8 = 0 in the interval [-10, 10]. The roots are approximately 0.9067 and 3.6888. From the given options, 0.91 is the closest to 0.9067. However, there is no option close to 3.6888. Let's analyze the original equation x 3 − 3 x 2 − 4 = x − 1 1 ​ + 5 . If x = 0.8 , then 0. 8 3 − 3 ( 0.8 ) 2 − 4 = 0.512 − 1.92 − 4 = − 5.408 . On the other hand, 0.8 − 1 1 ​ + 5 = − 0.2 1 ​ + 5 = − 5 + 5 = 0 . So x = 0.8 is not a solution.

Checking x=0.91 If x = 0.91 , then 0.9 1 3 − 3 ( 0.91 ) 2 − 4 = 0.753 − 2.484 − 4 = − 5.731 . On the other hand, 0.91 − 1 1 ​ + 5 = − 0.09 1 ​ + 5 = − 11.11 + 5 = − 6.11 . So x = 0.91 is not a solution.

Numerical Roots Let's try to find the roots numerically. The roots are approximately 0.907 and 3.79. Since we are given the options 0.91, -5.72, and 0.80, we can assume that 0.91 is one of the roots. However, none of the other options are close to 3.79. There must be an error in the question or the options.

Final Analysis Using the python_calculation_tool , we found the roots to be approximately 0.9067 and 3.6888. Given the options, the closest value to 0.9067 is 0.91. Since we need to select two values and the other options are -5.72 and 0.80, let's consider that there might be a typo in the problem and the second root is not among the options. Thus, we select 0.91 as one of the roots. However, we need to find another root from the given options. Since none of the options are close to 3.6888, we can try to plug in the values into the original equation to see which one is closer to being a solution. If we plug in 0.8, we get 0. 8 3 − 3 ( 0.8 ) 2 − 4 = − 5.408 and 0.8 − 1 1 ​ + 5 = 0 . If we plug in -5.72, we get ( − 5.72 ) 3 − 3 ( − 5.72 ) 2 − 4 = − 187.07 − 98.3 − 4 = − 289.37 and − 5.72 − 1 1 ​ + 5 = − 6.72 1 ​ + 5 = − 0.149 + 5 = 4.851 . Since none of the options are close to being a solution, and we know that 0.91 is close to one of the roots, we will select 0.91 as one of the roots and indicate that there might be an issue with the options provided.


Examples
When designing a bridge, engineers need to find the roots of complex equations to ensure the structure's stability. These roots represent critical points where forces balance, preventing collapse. Similarly, in electrical circuit design, finding the roots of equations helps determine the resonant frequencies, ensuring optimal performance and preventing damage to components. This problem demonstrates the importance of finding approximate solutions when exact solutions are difficult to obtain, a common scenario in many engineering and scientific applications.

Answered by GinnyAnswer | 2025-07-08