Multiple choice

The equation $\displaystyle \frac{(x + 2)(x - 5)}{(x - 3)(x + 6)}$ = $\displaystyle \frac{x - 2}{x +4}$ has

  1. $3$ roots
  2. $2$ roots
  3. $1$ root

  4. No root
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C Correct answer
Explanation

Cross-multiplying gives (x + 2)(x - 5)(x + 4) = (x - 2)(x - 3)(x + 6). Expanding both sides: (x^2 - 3x - 10)(x + 4) = x^3 + x^2 - 22x - 40; (x^2 - 5x + 6)(x + 6) = x^3 + x^2 - 24x + 36. Setting them equal: -22x - 40 = -24x + 36, which simplifies to 2x = 76, so x = 38. This is a single linear solution.

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