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Divide out all common factors (x+4)/(x-4)-(x-4)/(x+5)

Divide out all common factors\newlinex+4x−4−x−4x+5 \frac{x+4}{x-4}-\frac{x-4}{x+5}

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Full solution

Q. Divide out all common factors\newlinex+4x−4−x−4x+5 \frac{x+4}{x-4}-\frac{x-4}{x+5}
  1. Find Common Denominator: Combine the two fractions by finding a common denominator, which is (x−4)(x+5)(x-4)(x+5).
  2. Rewrite with Common Denominator: Rewrite each fraction with the common denominator: (x+4)(x+5)(x−4)(x+5)−(x−4)(x−4)(x−4)(x+5)\frac{(x+4)(x+5)}{(x-4)(x+5)} - \frac{(x-4)(x-4)}{(x-4)(x+5)}.
  3. Expand Numerators: Expand the numerators: (x2+5x+4x+20)/((x−4)(x+5))−(x2−4x−4x+16)/((x−4)(x+5))(x^2+5x+4x+20)/((x-4)(x+5)) - (x^2-4x-4x+16)/((x-4)(x+5)).
  4. Combine Like Terms: Combine like terms in the numerators: (x2+9x+20)/((x−4)(x+5))−(x2−8x+16)/((x−4)(x+5))(x^2+9x+20)/((x-4)(x+5)) - (x^2-8x+16)/((x-4)(x+5)).
  5. Subtract Numerators: Subtract the numerators over the common denominator: (x2+9x+20)−(x2−8x+16)(x−4)(x+5)\frac{(x^2+9x+20) - (x^2-8x+16)}{(x-4)(x+5)}.
  6. Final Simplification: Combine like terms in the numerator: (x2+9x+20−x2+8x−16)/((x−4)(x+5))(x^2+9x+20 - x^2+8x-16)/((x-4)(x+5)).

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