If the roots of the equation `bx^(2)+cx+a=0` be imaginary, then for all real values of x, the expression `3b^(2)x^(2)+6bcx +2c^(2)` is

Options

A. less than -4ab
B. greater than 4ab
C. less than 4ab
D. greater than -4ab

Answer

To solve the problem, we need to analyze the given quadratic equation and the expression provided. Let's break it down step by step.

### Step 1: Understanding the Condition for Imaginary Roots
The roots of the quadratic equation \( bx^2 + cx + a = 0 \) are imaginary if the discriminant \( D \) is less than zero. The discriminant is given by:
\[ D = c^2 - 4ab \]
For the roots to be imaginary, we have:
\[ c^2 - 4ab < 0 \]
This implies:
\[ c^2 < 4ab \]

### Step 2: Analyzing the Expression
We need to analyze the expression:
\[ 3b^2x^2 + 6bcx + 2c^2 \]
We can rewrite this expression by factoring out common terms.

### Step 3: Factoring the Expression
Notice that we can group the first two terms:
\[ 3b^2x^2 + 6bcx = 3(b^2x^2 + 2bcx) \]
Now, we can complete the square for \( b^2x^2 + 2bcx \):
\[ b^2x^2 + 2bcx = (bx + c)^2 - c^2 \]
Thus, we can rewrite the expression as:
\[ 3((bx + c)^2 - c^2) + 2c^2 = 3(bx + c)^2 - 3c^2 + 2c^2 = 3(bx + c)^2 - c^2 \]

### Step 4: Analyzing the Resulting Expression
Now we have:
\[ 3(bx + c)^2 - c^2 \]
Since \( (bx + c)^2 \) is always non-negative for all real \( x \), the minimum value of \( 3(bx + c)^2 \) is 0. Therefore, the expression simplifies to:
\[ 3(bx + c)^2 - c^2 \geq -c^2 \]
From our earlier analysis, we know:
\[ -c^2 > -4ab \quad \text{(since } c^2 < 4ab\text{)} \]
Thus, we can conclude:
\[ 3(bx + c)^2 - c^2 > -4ab \]

### Conclusion
Therefore, the expression \( 3b^2x^2 + 6bcx + 2c^2 \) is greater than \( -4ab \) for all real values of \( x \).

### Final Answer
\[ 3b^2x^2 + 6bcx + 2c^2 > -4ab \quad \text{for all real } x \]

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