Questions: -frac1x-1+frac2x-2=frac3x^2-3 x+2

$-frac1x-1+frac2x-2=frac3x^2-3 x+2$

Transcript text: \[ -\frac{1}{x-1}+\frac{2}{x-2}=\frac{3}{x^{2}-3 x+2} \]

Solution

Step 1: Factor the Denominator

We start with the equation

\[ -\frac{1}{x-1} + \frac{2}{x-2} = \frac{3}{x^2 - 3x + 2} \]

To find the least common denominator, we first factor the polynomial in the denominator on the right-hand side:

\[ x^2 - 3x + 2 = (x - 2)(x - 1) \]

Step 2: Rewrite the Left-Hand Side

Next, we rewrite the left-hand side of the equation using the least common denominator $(x - 1)(x - 2)$:

\[ -\frac{1}{x-1} + \frac{2}{x-2} = -\frac{1 \cdot (x - 2)}{(x - 1)(x - 2)} + \frac{2 \cdot (x - 1)}{(x - 1)(x - 2)} \]

This simplifies to:

\[ -\frac{x - 2}{(x - 1)(x - 2)} + \frac{2(x - 1)}{(x - 1)(x - 2)} = \frac{-x + 2 + 2x - 2}{(x - 1)(x - 2)} = \frac{x}{(x - 1)(x - 2)} \]

Step 3: Set Up the Equation

Now we can set the left-hand side equal to the right-hand side:

\[ \frac{x}{(x - 1)(x - 2)} = \frac{3}{(x - 1)(x - 2)} \]

Step 4: Equate the Numerators

Since the denominators are the same, we can equate the numerators:

\[ x = 3 \]

Step 5: Conclusion

The solution to the equation is $ x = 3 $.

$\boxed{3}$

Was this solution helpful?

Unhelpful

Helpful

Next >