Multiple choice

Solve the following quadratic equation by completing the square: $\dfrac{5x+7}{x-1}=3x+2$

  1. $\left \{ -1, 3  \right \}$
  2. $\left \{ 1, 3  \right \}$
  3. $\left \{ -1, -3  \right \}$

  4. None of these
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A Correct answer
Explanation

The equation simplifies to 5x + 7 = (3x + 2)(x - 1), which is 5x + 7 = 3x^2 - x - 2. Rearranging gives 3x^2 - 6x - 9 = 0, or x^2 - 2x - 3 = 0. Factoring yields (x - 3)(x + 1) = 0, so the roots are 3 and -1.

AI explanation

Multiply both sides of the equation by (x - 1) to get 5x + 7 = (x - 1)(3x + 2). Expanding and simplifying yields 5x + 7 = 3x^2 - x - 2, which rearranges to the quadratic equation 3x^2 - 6x - 9 = 0. Dividing the entire equation by 3 gives x^2 - 2x - 3 = 0, and factoring this provides (x - 3)(x + 1) = 0. The solutions for x are 3 and -1, so the roots are -1 and 3.

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