To find the value of [tex]\( x \)[/tex] that satisfies the equation
[tex]\[
\frac{x - 3}{x^2} + x - 2 = 0,
\][/tex]
we will solve this equation step-by-step.
1. Rewrite the equation:
[tex]\[
\frac{x - 3}{x^2} + x - 2 = 0.
\][/tex]
2. Combine all terms over a common denominator:
The common denominator for the terms is [tex]\( x^2 \)[/tex], hence:
[tex]\[
\frac{x - 3 + x^3 - 2x^2}{x^2} = 0.
\][/tex]
3. Simplify the numerator:
Combine like terms in the numerator:
[tex]\[
\frac{x^3 - 2x^2 + x - 3}{x^2} = 0.
\][/tex]
4. Set the numerator equal to zero:
Since the denominator [tex]\( x^2 \)[/tex] cannot be zero (because division by zero is undefined), we focus on the numerator:
[tex]\[
x^3 - 2x^2 + x - 3 = 0.
\][/tex]
5. Solve the polynomial equation:
We need to find the roots of the cubic polynomial:
[tex]\[
x^3 - 2x^2 + x - 3 = 0.
\][/tex]
The solutions to this cubic equation are:
[tex]\[
x = \frac{2}{3} + \left( -\frac{1}{2} - \frac{\sqrt{3}}{2}i \right) \left( \frac{\sqrt{77}}{6} + \frac{79}{54} \right)^{\frac{1}{3}} + \frac{1}{9 \left( -\frac{1}{2} - \frac{\sqrt{3}}{2}i \right) \left( \frac{\sqrt{77}}{6} + \frac{79}{54} \right)^{\frac{1}{3}}},
\][/tex]
[tex]\[
x = \frac{2}{3} + \frac{1}{9 \left( -\frac{1}{2} + \frac{\sqrt{3}}{2}i \right) \left( \frac{\sqrt{77}}{6} + \frac{79}{54} \right)^{\frac{1}{3}}} + \left( -\frac{1}{2} + \frac{\sqrt{3}}{2}i \right) \left( \frac{\sqrt{77}}{6} + \frac{79}{54} \right)^{\frac{1}{3}},
\][/tex]
[tex]\[
x = \frac{2}{3} + \left( \frac{\sqrt{77}}{6} + \frac{79}{54} \right)^{\frac{1}{3}} + \frac{1}{9 \left( \frac{\sqrt{77}}{6} + \frac{79}{54} \right)^{\frac{1}{3}}}.
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which the expression [tex]\(\frac{x - 3}{x^2} + x - 2 = 0\)[/tex] holds true are given by these complex solutions.
