To solve the given equation:
[tex]\[ x^3 - 3x^2 - 4 = \frac{1}{x - 1} + 5 \][/tex]
we first rewrite it in a single expression equal to zero:
[tex]\[ x^3 - 3x^2 - 4 - \frac{1}{x - 1} - 5 = 0 \][/tex]
This simplifies to:
[tex]\[ x^3 - 3x^2 - 9 - \frac{1}{x - 1} = 0 \][/tex]
To solve for [tex]\( x \)[/tex], we need to find the roots of this equation. These roots are the values of [tex]\( x \)[/tex] that satisfy the equation.
The approximate numerical solutions to this equation are:
1. [tex]\( x = 3.68875976307085 \)[/tex]
2. [tex]\( x = -0.297742382351758 - 1.51762970956905i \)[/tex] (a complex root)
3. [tex]\( x = -0.297742382351758 + 1.51762970956905i \)[/tex] (a complex root)
4. [tex]\( x = 0.906725001632668 \)[/tex]
Since we are typically interested in real solutions unless specified otherwise, the real solutions to this equation are approximately:
- [tex]\( x = 3.68875976307085 \)[/tex]
- [tex]\( x = 0.906725001632668 \)[/tex]
To plug into the drop-down menus:
The solutions to the equation are approximately [tex]\( x = \)[/tex] [tex]\(3.6888\)[/tex] (rounded to four decimal places) and [tex]\( x = \)[/tex] [tex]\(0.9067\)[/tex] (rounded to four decimal places).
