To solve the equation:
[tex]\[
\frac{2x}{x-1} - \frac{2x-5}{x^2 - 3x + 2} = \frac{-3}{x-2},
\][/tex]
let's follow these steps:
1. Factorize the Denominators:
Note that the quadratic expression [tex]\( x^2 - 3x + 2 \)[/tex] can be factorized:
[tex]\[
x^2 - 3x + 2 = (x-1)(x-2).
\][/tex]
2. Rewrite the Equation with Common Denominators:
Rewrite the equation with the quadratic denominator [tex]\( (x-1)(x-2) \)[/tex]:
[tex]\[
\frac{2x}{x-1} - \frac{2x-5}{(x-1)(x-2)} = \frac{-3}{x-2}
\][/tex]
To combine the fractions on the left-hand side, get a common denominator [tex]\((x-1)(x-2)\)[/tex]:
[tex]\[
\frac{2x(x-2)}{(x-1)(x-2)} - \frac{2x-5}{(x-1)(x-2)}.
\][/tex]
Expanding and combining:
[tex]\[
\frac{2x^2 - 4x - (2x - 5)}{(x-1)(x-2)},
\][/tex]
simplifying further:
[tex]\[
\frac{2x^2 - 4x - 2x + 5}{(x-1)(x-2)} = \frac{2x^2 - 6x + 5}{(x-1)(x-2)}.
\][/tex]
3. Equate the Fractions and Solve the Numerator Equation:
We now have:
[tex]\[
\frac{2x^2 - 6x + 5}{(x-1)(x-2)} = \frac{-3}{x-2}.
\][/tex]
Cross-multiplying to eliminate denominators:
[tex]\[
(2x^2 - 6x + 5)(x-2) = -3(x-1),
\][/tex]
simplifying:
[tex]\[
2x^3 - 4x^2 - 6x^2 + 12x + 5x - 10 = -3x + 3,
\][/tex]
[tex]\[
2x^3 - 10x^2 + 17x - 10 = 3.
\][/tex]
Combine like terms on one side:
[tex]\[
2x^3 - 10x^2 + 17x - 13 = 0.
\][/tex]
4. Factoring and Finding Roots:
Now we need to find the roots of the polynomial equation [tex]\( 2x^3 - 10x^2 + 17x - 13 = 0 \)[/tex].
Solving this cubic equation, we find the complex roots:
[tex]\[
x = 0.75 - 0.661437827766148i, \quad x = 0.75 + 0.661437827766148i.
\][/tex]
Checking our solutions, we exclude [tex]\( x=1 \)[/tex] and [tex]\( x=2 \)[/tex] as they make the denominators zero, thus verifying our original restrictions.
Therefore, the valid solutions are [tex]\( x = 0.75 - 0.661437827766148i \)[/tex] and [tex]\( x = 0.75 + 0.661437827766148i \)[/tex], which are complex numbers.
Thus, none of the options A, B, C, or D given (which suggest real solutions) are correct based on the problem statement. The equation has two complex solutions.
