Multiple choice

If the roots of the equation $\displaystyle bx^{2}+cx+a= 0$ be imaginary, then for all real values of $x$. The expression $\displaystyle 3b^{2}x^{2}+6bcx+2c^{2}$ is

  1. less than $4ab$
  2. greater than $-4ab$
  3. less than $-4ab$
  4. greater than $4ab$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If roots of bx^2 + cx + a = 0 are imaginary, the discriminant c^2 - 4ab < 0, so c^2 < 4ab. The expression 3b^2x^2 + 6bcx + 2c^2 can be written as 3(bx + c)^2 - c^2. Since c^2 < 4ab, -c^2 > -4ab. This logic is complex, but testing values confirms the inequality.

AI explanation

Because the roots of bx^2 + cx + a = 0 are imaginary, its discriminant is negative, meaning c^2 - 4ab < 0 and therefore 4ab > c^2. We want to find the minimum value of the expression E = 3b^2x^2 + 6bcx + 2c^2 for real x, which is a parabola opening upwards. The minimum occurs at x = -6bc / (6b^2) = -c/b, and substituting this back gives E_min = 3b^2(-c/b)^2 + 6bc(-c/b) + 2c^2 = 3c^2 - 6c^2 + 2c^2 = -c^2. Since 4ab > c^2, we have -4ab < -c^2, which means the expression is always greater than -4ab.

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