Multiple choice

The equation $(a+2){x}^{2}+(a-3)x=2a-1,a\neq -2$ has rational roots for

  1. all rational values of $a$ except $a=-2$
  2. all real values of $a$ except $a=-2$
  3. rational values of $a> \displaystyle \frac{1}{2}$

  4. none of these
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A Correct answer
Explanation

For the roots of a quadratic Ax^2 + Bx + C = 0 to be rational, the discriminant D = B^2 - 4AC must be a perfect square. Here, D = (a - 3)^2 - 4(a + 2)(-(2a - 1)) = (a - 3)^2 + 4(a + 2)(2a - 1) = a^2 - 6a + 9 + 4(2a^2 + 3a - 2) = 9a^2 + 6a + 1 = (3a + 1)^2. Since the discriminant is a perfect square for any rational a, the roots are rational.

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