The given equation is:

$\frac{1}{x^2 - 3x + 2} = \frac{3}{x - 1} - \frac{3}{x - 2} + \frac{5}{x^2 - 4x + 4}.$

Step-by-Step Solution:

Step 1: Factorize the denominators wherever possible.

1. $x^2 - 3x + 2$: Factorize: $x^2 - 3x + 2 = (x - 1)(x - 2).$

   So: $\frac{1}{x^2 - 3x + 2} = \frac{1}{(x - 1)(x - 2)}.$

2. $x^2 - 4x + 4$: Factorize: $x^2 - 4x + 4 = (x - 2)^2.$

   So: $\frac{5}{x^2 - 4x + 4} = \frac{5}{(x - 2)^2}.$

The equation now becomes: $\frac{1}{(x - 1)(x - 2)} = \frac{3}{x - 1} - \frac{3}{x - 2} + \frac{5}{(x - 2)^2}.$

Step 2: Find the least common denominator (LCD).

The denominators are:

1. $(x - 1)(x - 2)$,
2. $(x - 1)$,
3. $(x - 2)$,
4. $(x - 2)^2$.

The LCD is: $(x - 1)(x - 2)^2.$

Step 3: Rewrite each term with the LCD.

1. For $\frac{1}{(x - 1)(x - 2)}$: Multiply numerator and denominator by $(x - 2)$: $\frac{1}{(x - 1)(x - 2)} = \frac{x - 2}{(x - 1)(x - 2)^2}.$

2. For $\frac{3}{x - 1}$: Multiply numerator and denominator by $(x - 2)^2$: $\frac{3}{x - 1} = \frac{3(x - 2)^2}{(x - 1)(x - 2)^2}.$

3. For $\frac{3}{x - 2}$: Multiply numerator and denominator by $(x - 1)(x - 2)$: $\frac{3}{x - 2} = \frac{3(x - 1)}{(x - 1)(x - 2)^2}.$

4. For $\frac{5}{(x - 2)^2}$: Multiply numerator and denominator by $(x - 1)$: $\frac{5}{(x - 2)^2} = \frac{5(x - 1)}{(x - 1)(x - 2)^2}.$

Step 4: Combine everything into one fraction.

The equation becomes: $\frac{x - 2}{(x - 1)(x - 2)^2} = \frac{3(x - 2)^2}{(x - 1)(x - 2)^2} - \frac{3(x - 1)}{(x - 1)(x - 2)^2} + \frac{5(x - 1)}{(x - 1)(x - 2)^2}.$

Combine all numerators over the common denominator $(x - 1)(x - 2)^2$: $x - 2 = 3(x - 2)^2 - 3(x - 1) + 5(x - 1).$

Step 5: Expand and simplify the numerators.

1. Expand $3(x - 2)^2$: $3(x - 2)^2 = 3(x^2 - 4x + 4) = 3x^2 - 12x + 12.$

2. Expand $-3(x - 1)$: $-3(x - 1) = -3x + 3.$

3. Expand $5(x - 1)$: $5(x - 1) = 5x - 5.$

Combine these: $3(x - 2)^2 - 3(x - 1) + 5(x - 1) = 3x^2 - 12x + 12 - 3x + 3 + 5x - 5.$

Simplify: $3x^2 - 12x + 12 - 3x + 3 + 5x - 5 = 3x^2 - 10x + 10.$

The equation now becomes: $x - 2 = 3x^2 - 10x + 10.$

Step 6: Rearrange into standard quadratic form.

Bring all terms to one side: $0 = 3x^2 - 10x + 10 - (x - 2).$

Simplify: $0 = 3x^2 - 10x + 10 - x + 2.$

$0 = 3x^2 - 11x + 12.$

Step 7: Solve the quadratic equation.

The equation is: $3x^2 - 11x + 12 = 0.$

Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$ where $a = 3$, $b = -11$, $c = 12$.

1. Calculate the discriminant: $\Delta = b^2 - 4ac = (-11)^2 - 4(3)(12) = 121 - 144 = -23.$

Since the discriminant is negative ($\Delta = -23$), there are no real solutions.

Final Answer:

The equation has no real solutions.

Do you have any questions, or would you like additional details on any step?

Related Questions:

1. How do you solve quadratic equations with a negative discriminant?
2. Can this equation have complex solutions? If so, how would they be calculated?
3. What is the significance of the least common denominator (LCD) in solving rational equations?
4. How do you factorize higher-degree polynomials?
5. What are some real-life applications of solving equations like these?

Tip:

Always check for factorization opportunities before solving equations—it simplifies the process significantly!