Multiple choice

Find the roots of the following equation. $\displaystyle \frac{1}{x +4} - \frac{1}{x - 7} = \frac{11}{30}, x \neq {-4 , 7}$

  1. $-1,-2$
  2. $-1,-3$
  3. $1,3$
  4. $1,2$
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D Correct answer
Explanation

Simplifying the left side of the equation gives -11 / (x^2 - 3x - 28) = 11 / 30. Dividing both sides by 11 and cross-multiplying yields x^2 - 3x - 28 = -30, which simplifies to the quadratic equation x^2 - 3x + 2 = 0. Factoring this equation gives (x - 1)(x - 2) = 0, so the roots are 1 and 2.

AI explanation

Multiplying the entire equation by 30(x + 4)(x - 7) eliminates the fractions, giving 30(x - 7) - 30(x + 4) equals 11(x + 4)(x - 7). Simplifying the left side yields negative 330, and expanding the right side gives 11(x squared - 3x - 28), which simplifies to 11x squared - 33x - 308. Bringing all terms to one side results in the quadratic equation 11x squared - 33x + 22 equals 0, which simplifies by dividing by 11 to x squared - 3x + 2 equals 0. Factoring this gives (x - 1)(x - 2) equals 0, so the roots are 1 and 2. The roots are 1 and 2.

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