Multiple choice

Solve the following quadratic equations by factorization method: $2(x+1)^2-5(x+1)=12$

  1. $\left \{ -\dfrac{5}{2}, 3 \right \}$ 
  2. $\left \{ \dfrac{5}{2}, 3 \right \}$ 
  3. $\left \{ -\dfrac{5}{2}, -3 \right \}$ 

  4. None of these
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A Correct answer
Explanation

Let u = x+1. The equation becomes 2u^2 - 5u - 12 = 0. Factoring: 2u^2 - 8u + 3u - 12 = 0, 2u(u-4) + 3(u-4) = 0, (2u+3)(u-4) = 0. So u = -3/2 or u = 4. Since u = x+1, x = -3/2 - 1 = -5/2 or x = 4 - 1 = 3.

AI explanation

Let y = x + 1 to simplify the equation to 2y^2 - 5y = 12, which becomes 2y^2 - 5y - 12 = 0. By splitting the middle term, we get 2y^2 - 8y + 3y - 12 = 0, which factors as (2y + 3)(y - 4) = 0. This gives y = -3/2 or y = 4, meaning x + 1 = -3/2 or x + 1 = 4. Solving for x gives the values -5/2 and 3, making the solution set -5/2, 3.

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