Multiple choice

If the equation $cx^{2} + bx - 2a = 0$ has no real roots and $a < \dfrac{b+c}{2}$, then choose the wrong option.

  1. $ac < 0$
  2. $a < 0$
  3. $\dfrac{c-b}{2}> a$
  4. $\dfrac{c+2b}{8}<a$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For no real roots, discriminant b^2 - 4ac < 0. Option D is the incorrect statement based on the properties of the quadratic equation given.

AI explanation

For the quadratic equation cx^2 + bx - 2a = 0 to have no real roots its discriminant must be negative meaning b^2 - 4(c)(-2a) < 0 or b^2 + 8ac < 0. This requires c to be non-zero and forces ac < 0 making option A correct. The condition a < (b + c) / 2 rearranges to 2a - b - c < 0. Examining (c + 2b) / 8 < a we can test a case where a = -1 b = 0 and c = 1 which satisfies b^2 + 8ac = -8 < 0 and -1 < (0 + 1) / 2. However (1 + 0) / 8 < -1 evaluates to 0.125 < -1 which is false. Therefore the wrong option is (c + 2b) / 8 < a.

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