Multiple choice

Find the roots of the following quadratic equations by using the quadratic formula $\displaystyle 2 \left( \frac{x}{x + 1} \right)^2 - 5 \left( \frac{x}{x + 1} \right) + 2 = 0, x \neq - 1$

  1. $1, 2$
  2. $1, -\cfrac{1}{2}$
  3. $\cfrac{1}{2},-2$
  4. $-2, 1$
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D Correct answer
Explanation

Let y = x / (x + 1). The equation is 2y^2 - 5y + 2 = 0. Factoring gives (2y - 1)(y - 2) = 0, so y = 1/2 or y = 2. If x/(x+1) = 1/2, then 2x = x + 1, so x = 1. If x/(x+1) = 2, then x = 2x + 2, so x = -2.

AI explanation

Substituting y for the expression x divided by (x plus 1), the equation becomes 2y squared minus 5y plus 2 equals 0. Factoring this quadratic yields (2y minus 1) times (y minus 2) equals 0, giving y equals one half or y equals 2. Substituting back gives x divided by (x plus 1) equals one half, which simplifies to x equals 1, and x divided by (x plus 1) equals 2, which simplifies to x equals negative 2. The roots are negative 2 and 1.

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