Key Concepts and Formulas

• Discriminant of a Quadratic: For a quadratic equation $Ax^2 + Bx + C = 0$, the roots are imaginary if the discriminant $D = B^2 - 4AC < 0$.
• Completing the Square: A technique used to rewrite a quadratic expression in the form $a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• Properties of Inequalities: Multiplying an inequality by a negative number reverses the inequality sign. If $a < b$ and $b < c$, then $a < c$.

Step-by-Step Solution

Step 1: Establish the condition for imaginary roots

We are given that the roots of the equation $bx^2 + cx + a = 0$ are imaginary. For this quadratic equation, the discriminant must be negative. $D = c^2 - 4(b)(a) < 0$ $c^2 - 4ab < 0$ $c^2 < 4ab$ This inequality is crucial for relating $a$, $b$, and $c$. It implies that $4ab$ is a positive quantity since $c^2 \ge 0$. WHY: This sets up the fundamental relationship between the coefficients based on the problem statement, which we'll use later.

Step 2: Analyze the expression $E(x)$

We are asked to find the range of the expression $E(x) = 3b^2x^2 + 6bcx + 2c^2$ for all real values of $x$. This is a quadratic expression in $x$. The coefficient of $x^2$ is $3b^2$. Since the roots of $bx^2+cx+a=0$ are imaginary, $b \neq 0$, so $b^2 > 0$, and $3b^2 > 0$. Thus, the parabola opens upward, meaning $E(x)$ has a minimum value. WHY: Recognizing $E(x)$ as an upward-opening parabola allows us to focus on finding its minimum value to determine its range.

Step 3: Find the minimum value of $E(x)$ by completing the square

To find the minimum value, we complete the square for $E(x)$: $E(x) = 3b^2x^2 + 6bcx + 2c^2$ $E(x) = 3b^2\left(x^2 + \frac{2c}{b}x\right) + 2c^2$ To complete the square inside the parentheses, we add and subtract $\left(\frac{c}{b}\right)^2$: $E(x) = 3b^2\left(x^2 + \frac{2c}{b}x + \left(\frac{c}{b}\right)^2 - \left(\frac{c}{b}\right)^2\right) + 2c^2$ $E(x) = 3b^2\left(\left(x + \frac{c}{b}\right)^2 - \frac{c^2}{b^2}\right) + 2c^2$ $E(x) = 3b^2\left(x + \frac{c}{b}\right)^2 - 3c^2 + 2c^2$ $E(x) = 3b^2\left(x + \frac{c}{b}\right)^2 - c^2$ Since $3b^2\left(x + \frac{c}{b}\right)^2 \ge 0$ for all real $x$, the minimum value of $E(x)$ occurs when $x = -\frac{c}{b}$. The minimum value is: $E_{min} = -c^2$ Therefore, $E(x) \ge -c^2$ for all real $x$. WHY: Completing the square transforms the quadratic into a form that reveals its minimum value, which is crucial for determining its lower bound.

Step 4: Use the condition $c^2 < 4ab$ to relate the minimum value to $4ab$

We know that $c^2 < 4ab$. Multiplying by -1, we get: $-c^2 > -4ab$ Since $E(x) \ge -c^2$, we can substitute $-c^2$ with something that is less than it, so we can say: $E(x) > -4ab$ While we know $E(x) > -4ab$, this doesn't directly help us prove that $E(x) < 4ab$. We want to show that $E(x)$ is always less than $4ab$. Let's rewrite $E(x)$ as follows: $E(x) = 3b^2 x^2 + 6bcx + 2c^2$ Let's consider $4ab - E(x) = 4ab - (3b^2 x^2 + 6bcx + 2c^2)$. We want to show that this is always positive. $4ab - E(x) = 4ab - 3b^2 x^2 - 6bcx - 2c^2$ The expression is a quadratic in $x$, and it has a maximum value when $x = - \frac{6bc}{2(-3b^2)} = \frac{c}{b}$. Plugging this in, we get $4ab - E(x) = 4ab - 3b^2 \frac{c^2}{b^2} + 6bc \frac{c}{b} - 2c^2 = 4ab - c^2$ Since $c^2 < 4ab$, $4ab - c^2 > 0$. We also know that the coefficient of the $x^2$ term, $-3b^2$, is negative, so the expression $4ab - E(x)$ is a downward-facing parabola. Its maximum value is $4ab - c^2$, which we know is positive. Therefore, $4ab - E(x)$ is always positive, meaning $E(x) < 4ab$ for all $x$. WHY: This step is crucial to establishing the upper bound of $E(x)$ and linking it to the given condition. By analyzing $4ab - E(x)$, we show that it is always positive, thus proving that $E(x) < 4ab$.

Common Mistakes & Tips

• Sign Errors: Be very careful with signs when completing the square and manipulating inequalities. A simple sign error can lead to an incorrect result.
• Confusing Minimum and Maximum: Understand the shape of the parabola to correctly identify whether you are finding a minimum or maximum value.
• Misinterpreting Imaginary Roots: Remember that imaginary roots imply the discriminant is strictly less than zero.

Summary

We were given that the roots of $bx^2 + cx + a = 0$ are imaginary, implying $c^2 < 4ab$. We analyzed the expression $E(x) = 3b^2x^2 + 6bcx + 2c^2$ by completing the square and found that its minimum value is $-c^2$. By analyzing the expression $4ab - E(x)$, we showed that it is always positive and $E(x) < 4ab$. Therefore, for all real values of $x$, the expression $3b^2x^2 + 6bcx + 2c^2$ is less than $4ab$.

Final Answer

The final answer is \boxed{less than 4ab}, which corresponds to option (A).