Multiple choice

If the roots of the equation $b x ^ { 2 } + c x + a = 0$ be imaginary, then for all real values of $x ,$ the expression $3 b ^ { 2 } x ^ { 2 } + 6 b c x + 2 c ^ { 2 }$ is -

  1. Greater than $- 4 a b$
  2. Less than $- 4 a b$
  3. Greater than $4\mathrm { ab }$
  4. Less than $4\mathrm { ab }$
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A Correct answer
Explanation

If bx^2 + cx + a = 0 has imaginary roots, 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. Thus, the expression is greater than -4ab.

AI explanation

Let the discriminant of the roots be negative, meaning $c^2 - 4ab < 0$, so $c^2 < 4ab$. Using the method of completing the square, we rewrite the expression as $3(bx + c)^2 - c^2$. We want to find the minimum value of this expression, which occurs when the squared term is zero, leaving a minimum of $-c^2$. Since $c^2 < 4ab$, multiplying by -1 reverses the inequality to give $-c^2 > -4ab$. Therefore, the entire expression is always greater than $-4ab$.

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