To solve the equation [tex]\( x - 210 + x = x^3 \)[/tex], let’s first simplify this equation step-by-step.
Given:
[tex]\[ x - 210 + x = x^3 \][/tex]
Simplify the left side by combining like terms:
[tex]\[ 2x - 210 = x^3 \][/tex]
Rearrange the equation to set it to zero:
[tex]\[ x^3 - 2x + 210 = 0 \][/tex]
We need to solve [tex]\( x^3 - 2x + 210 = 0 \)[/tex] for [tex]\( x \)[/tex]. The solutions to this cubic equation are quite complex, and they involve complex numbers and roots. These solutions are:
[tex]\[ x_1 = -\frac{2}{\left(-\frac{1}{2} - \frac{\sqrt{3}}{2}i\right)\left(3\sqrt{893001} + 2835\right)^{1/3}} - \frac{\left(-\frac{1}{2} - \frac{\sqrt{3}}{2}i\right)\left(3\sqrt{893001} + 2835\right)^{1/3}}{3} \][/tex]
[tex]\[ x_2 = -\frac{\left(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)\left(3\sqrt{893001} + 2835\right)^{1/3}}{3} - \frac{2}{\left(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)\left(3\sqrt{893001} + 2835\right)^{1/3}} \][/tex]
[tex]\[ x_3 = -\frac{\left(3\sqrt{893001} + 2835\right)^{1/3}}{3} - \frac{2}{\left(3\sqrt{893001} + 2835\right)^{1/3}} \][/tex]
These solutions are denoted as [tex]\( x_1 \)[/tex], [tex]\( x_2 \)[/tex], and [tex]\( x_3 \)[/tex].
Next, we calculate [tex]\( 23x + 4 \)[/tex] for each solution.
For [tex]\( x_1 \)[/tex]:
[tex]\[ 23x_1 + 4 = 4 - \frac{46}{\left( -\frac{1}{2} - \frac{\sqrt{3}}{2}i \right) \left( 3\sqrt{893001} + 2835 \right)^{1/3}} - \frac{23\left( -\frac{1}{2} - \frac{\sqrt{3}}{2}i \right) \left( 3\sqrt{893001} + 2835 \right)^{1/3}}{3} \][/tex]
For [tex]\( x_2 \)[/tex]:
[tex]\[ 23x_2 + 4 = 4 - \frac{23\left( -\frac{1}{2} + \frac{\sqrt{3}}{2}i \right) \left( 3\sqrt{893001} + 2835 \right)^{1/3}}{3} - \frac{46}{\left( -\frac{1}{2} + \frac{\sqrt{3}}{2}i \right) \left( 3\sqrt{893001} + 2835 \right)^{1/3}} \][/tex]
For [tex]\( x_3 \)[/tex]:
[tex]\[ 23x_3 + 4 = -\frac{23 \left( 3\sqrt{893001} + 2835 \right)^{1/3}}{3} - \frac{46}{\left( 3\sqrt{893001} + 2835 \right)^{1/3}} + 4 \][/tex]
This is the detailed solution for the values of [tex]\( 23x + 4 \)[/tex] given the solutions of the cubic equation [tex]\( x^3 - 2x + 210 = 0 \)[/tex].
