If the roots of the equation bx² + cx + a = 0 be imaginary, then for all real values of x, the expre

Q: Wait, why do we reject the negative root here? Or why is the discriminant less than zero?

Great question! When we say a quadratic equation has imaginary roots, it means the roots are not real numbers. In the quadratic formula $x = \frac{-b \pm \sqrt{D}}{2a}$, the term under the square root, $D = b^2 - 4ac$, determines the nature of the roots. If $D < 0$, we are taking the square root of a negative number, which results in imaginary values. That is why we set the discriminant to be strictly less than zero.

Q: Why can't we just use energy conservation or some other physics principle?

I love the enthusiasm! However, this is a pure algebraic inequality problem. We are dealing with the properties of quadratic functions. The 'physics' here is the geometry of the parabola. By understanding that the expression is an upward-opening parabola, we can find its minimum value, which acts as a 'floor' for the entire expression. It is like finding the lowest point of a valley!

Q: Why did the inequality sign flip when I multiplied by $-1$?

That is a classic trap! Remember the fundamental rule of inequalities: whenever you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must reverse. For example, $2 -5$ is the correct statement. Always keep this in mind during your JEE preparation!

Question

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

Options:

(A) less than $4ab$

(B) greater than $- 4ab$

(C) less than $- 4ab$

(D) greater than $4ab$

Sigma Percentile · Accepted Answer

### Analyzing the Setup Imagine you are standing on a vast, flat plain, and before you lies a quadratic equation: $bx^2 + cx + a = 0$. You are told that its roots are imaginary. In the world of real numbers, this means the graph of this quadratic never touches the $x$-axis; it floats entirely above or below it. Mathematically, this is captured by the discriminant, $D = c^2 - 4ab$. For the roots to be imaginary, we must have $D < 0$. This gives us our first vital clue: $c^2 < 4ab$. Keep this inequality close; it is the key that will unlock the final door. ### The Target Expression Now, shift your focus to the expression we need to conquer: $f(x) = 3b^2x^2 + 6bcx + 2c^2$. This is a quadratic expression in $x$. To understand its behavior, we look at its leading coefficient, $A = 3b^2$. Since $b$ is a real number and $b eq 0$, $b^2$ is always positive, which means $3b^2 > 0$. Geometrically, this tells us that the graph of $f(x)$ is a parabola that opens upwards. It has a bottom, a minimum point, and all other values lie above it. ### The Search for the Minimum To find the range of this expression, we need to find its minimum value. For any quadratic $Ax^2 + Bx + C$, the minimum value is given by the formula: $$-\frac{D'}{4A}$$ Here, $D'$ is the discriminant of that specific quadratic. Let us calculate $D'$ for our expression $f(x) = 3b^2x^2 + 6bcx + 2c^2$, where $A = 3b^2$, $B = 6bc$, and $C = 2c^2$. Plugging these into the discriminant formula $D' = B^2 - 4AC$: $$D' = (6bc)^2 - 4(3b^2)(2c^2)$$ Expanding this, we find: $$D' = 36b^2c^2 - 24b^2c^2 = 12b^2c^2$$ ### The Final Connection Now, we calculate the minimum value: $$ ext{Min} = -\frac{12b^2c^2}{4(3b^2)}$$ The $12b^2$ terms in the numerator and denominator cancel out with satisfying precision, leaving us with just $-c^2$. So, we know that for all real $x$: $$f(x) \ge -c^2$$ But remember our first clue? We know $c^2 < 4ab$. If we multiply this inequality by $-1$, the sign flips, giving us $-c^2 > -4ab$. By the transitive property, since $f(x) \ge -c^2$ and $-c^2 > -4ab$, it must be true that **$f(x) > -4ab$**. We have navigated the geometry of the parabola and the logic of inequalities to arrive at the truth.

Animated Solution for Mathematics - Quadratic Equations: 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

Select Answer:

Visualized Solution

Visual Anchor & Problem Statement

The Discriminant Condition

Introduce the Target Expression

Analyze the Target Quadratic

Logic Bridge - Minimum Value Formula

Raw Setup - Calculate $D^{'}$

Atomic Compute - Simplify $D^{'}$

Atomic Compute - Minimum Value

The Way Forward - Connect with Step 2

Atomic Compute - Inequality Manipulation

Final Conclusion

Summary

The Sigma Insight: Maximum and Minimum Values of Quadratic Expressions

Analyzing the Setup

The Target Expression

The Search for the Minimum

The Final Connection

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Animated Solution for Mathematics - Quadratic Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the roots of the equation bx2+cx+a=0 be imaginary, then for all real values of x, the expression 3b2x2+6bcx+2c2 is