Multiple choice

The quadratic equations $2x^{2} - (a^{3} + 8a - 1)x + a^{2} - 4a = 0$ possesses roots of opposite sign. Then

  1. $a\leq 0$
  2. $0 < a < 4$
  3. $4\leq a < 8$
  4. $a \geq 8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For roots to have opposite signs, the product of roots (c/a) must be negative. Here, (a^2 - 4a) / 2 < 0. So a(a - 4) < 0. This inequality holds when 0 < a < 4.

AI explanation

For the quadratic equation 2x^2 - (a^3 + 8a - 1)x + a^2 - 4a = 0 to have roots of opposite sign, the product of its roots must be negative. By Vieta's formulas, the product of the roots equals c/a, so (a^2 - 4a)/2 must be less than zero. Multiplying by 2 gives a(a - 4) < 0, which implies 0 < a < 4.

Keep practicing — related questions