Question

\frac{x+3}{x^{2}-2x+1} + \frac{x}{x^{2}-3x+2}
\frac{x}{x^{2}-4x+4} - \frac{2}{x^{2}-4}

Explanation:

First Expression: $\boldsymbol{\frac{x + 3}{x^2 - 2x + 1} + \frac{x}{x^2 - 3x + 2}}$

Step 1: Factor Denominators

Factor $x^2 - 2x + 1$: It's a perfect square, so $x^2 - 2x + 1=(x - 1)^2$.
Factor $x^2 - 3x + 2$: Find two numbers that multiply to $2$ and add to $-3$, which are $-1$ and $-2$. So $x^2 - 3x + 2=(x - 1)(x - 2)$.
Now the expression is $\frac{x + 3}{(x - 1)^2} + \frac{x}{(x - 1)(x - 2)}$.

Step 2: Find Common Denominator

The common denominator (LCD) of $(x - 1)^2$ and $(x - 1)(x - 2)$ is $(x - 1)^2(x - 2)$.

Step 3: Rewrite Fractions with LCD

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

Now the expression is $\frac{(x + 3)(x - 2) + x(x - 1)}{(x - 1)^2(x - 2)}$.

Step 4: Expand and Simplify Numerator

Expand $(x + 3)(x - 2)$: $x^2 - 2x + 3x - 6 = x^2 + x - 6$.
Expand $x(x - 1)$: $x^2 - x$.
Add the two expanded numerators: $(x^2 + x - 6) + (x^2 - x) = 2x^2 - 6$.
Factor the numerator: $2x^2 - 6 = 2(x^2 - 3) = 2(x - \sqrt{3})(x + \sqrt{3})$ (or leave as $2x^2 - 6$).

So the simplified form is $\frac{2x^2 - 6}{(x - 1)^2(x - 2)}$ (or $\frac{2(x^2 - 3)}{(x - 1)^2(x - 2)}$).

Second Expression: $\boldsymbol{\frac{x}{x^2 - 4x + 4} - \frac{2}{x^2 - 4}}$

Step 1: Factor Denominators

Factor $x^2 - 4x + 4$: Perfect square, so $x^2 - 4x + 4=(x - 2)^2$.
Factor $x^2 - 4$: Difference of squares, so $x^2 - 4=(x - 2)(x + 2)$.
Now the expression is $\frac{x}{(x - 2)^2} - \frac{2}{(x - 2)(x + 2)}$.

Step 2: Find Common Denominator

The LCD of $(x - 2)^2$ and $(x - 2)(x + 2)$ is $(x - 2)^2(x + 2)$.

Step 3: Rewrite Fractions with LCD

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

Now the expression is $\frac{x(x + 2) - 2(x - 2)}{(x - 2)^2(x + 2)}$.

Step 4: Expand and Simplify Numerator

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

So the simplified form is $\frac{x^2 + 4}{(x - 2)^2(x + 2)}$.

Final Answers:

• First expression: $\boldsymbol{\frac{2x^2 - 6}{(x - 1)^2(x - 2)}}$ (or $\boldsymbol{\frac{2(x^2 - 3)}{(x - 1)^2(x - 2)}}$)
• Second expression: $\boldsymbol{\frac{x^2 + 4}{(x - 2)^2(x + 2)}}$

Answer:

Step 1: Factor Denominators

Factor $x^2 - 4x + 4$: Perfect square, so $x^2 - 4x + 4=(x - 2)^2$.
Factor $x^2 - 4$: Difference of squares, so $x^2 - 4=(x - 2)(x + 2)$.
Now the expression is $\frac{x}{(x - 2)^2} - \frac{2}{(x - 2)(x + 2)}$.

Step 2: Find Common Denominator

The LCD of $(x - 2)^2$ and $(x - 2)(x + 2)$ is $(x - 2)^2(x + 2)$.

Step 3: Rewrite Fractions with LCD

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

Now the expression is $\frac{x(x + 2) - 2(x - 2)}{(x - 2)^2(x + 2)}$.

Step 4: Expand and Simplify Numerator

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

So the simplified form is $\frac{x^2 + 4}{(x - 2)^2(x + 2)}$.

Final Answers:

• First expression: $\boldsymbol{\frac{2x^2 - 6}{(x - 1)^2(x - 2)}}$ (or $\boldsymbol{\frac{2(x^2 - 3)}{(x - 1)^2(x - 2)}}$)
• Second expression: $\boldsymbol{\frac{x^2 + 4}{(x - 2)^2(x + 2)}}$