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QUESTION IMAGE

\\(\\frac{x + 3}{x^2 - 2x + 1} + \\frac{x}{x^2 - 3x + 2}\\) \\(\\frac{x…

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:

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

Step 1: Factor Denominators

Factor each quadratic denominator:

  • $x^2 - 2x + 1 = (x - 1)^2$ (perfect square trinomial).
  • $x^2 - 3x + 2 = (x - 1)(x - 2)$ (factor by finding two numbers that multiply to 2 and add to -3).

Step 2: Find LCD

The least common denominator (LCD) is $(x - 1)^2(x - 2)$ (product of the highest powers of all distinct factors).

Step 3: Rewrite Fractions with LCD

  • $\frac{x + 3}{(x - 1)^2} = \frac{(x + 3)(x - 2)}{(x - 1)^2(x - 2)}$ (multiply numerator/denominator by $(x - 2)$).
  • $\frac{x}{(x - 1)(x - 2)} = \frac{x(x - 1)}{(x - 1)^2(x - 2)}$ (multiply numerator/denominator by $(x - 1)$).

Step 4: Add Numerators

Combine the fractions:
$$\frac{(x + 3)(x - 2) + x(x - 1)}{(x - 1)^2(x - 2)}$$

Expand the numerators:

  • $(x + 3)(x - 2) = x^2 + x - 6$
  • $x(x - 1) = x^2 - x$

Add them: $x^2 + x - 6 + x^2 - x = 2x^2 - 6$

Simplify numerator: $2x^2 - 6 = 2(x^2 - 3)$

So, the first expression simplifies to:
$$\frac{2(x^2 - 3)}{(x - 1)^2(x - 2)}$$

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

Step 1: Factor Denominators

Factor each quadratic denominator:

  • $x^2 - 4x + 4 = (x - 2)^2$ (perfect square trinomial).
  • $x^2 - 4 = (x - 2)(x + 2)$ (difference of squares: $a^2 - b^2 = (a - b)(a + b)$).

Step 2: Find LCD

The LCD is $(x - 2)^2(x + 2)$ (product of the highest powers of all distinct factors).

Step 3: Rewrite Fractions with LCD

  • $\frac{x}{(x - 2)^2} = \frac{x(x + 2)}{(x - 2)^2(x + 2)}$ (multiply numerator/denominator by $(x + 2)$).
  • $\frac{2}{(x - 2)(x + 2)} = \frac{2(x - 2)}{(x - 2)^2(x + 2)}$ (multiply numerator/denominator by $(x - 2)$).

Step 4: Subtract Numerators

Combine the fractions:
$$\frac{x(x + 2) - 2(x - 2)}{(x - 2)^2(x + 2)}$$

Expand the numerators:

  • $x(x + 2) = x^2 + 2x$
  • $2(x - 2) = 2x - 4$

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

So, the second expression simplifies to:
$$\frac{x^2 + 4}{(x - 2)^2(x + 2)}$$

Answer:

s:

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