Problem 153 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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 (A) greater than \(4 a b\) (B) less than \(4 a b\) (C) greater than \(-4 a b\) (D) less than \(-4 a b\)

Step-by-Step Solution

Answer

(A) greater than 4ab.

Step 1: Understand Imaginary Roots

For the roots of a quadratic equation to be imaginary, the discriminant must be less than zero. The equation given is \(bx^2 + cx + a = 0\). The discriminant \(\Delta\) of this quadratic is \(\Delta = c^2 - 4ab\). Since the roots are imaginary, we have \(c^2 - 4ab < 0\).

Step 2: Analyze the New Expression

We need to analyze the expression \(3b^2x^2 + 6bcx + 2c^2\) using the condition derived in Step 1. This expression is a quadratic in \(x\) with \(A = 3b^2\), \(B = 6bc\), and \(C = 2c^2\).

Step 3: Investigate Sum and Product of Roots

For any quadratic \(Ax^2 + Bx + C\), the sum of roots \(\alpha + \beta = -\frac{B}{A}\) and product \(\alpha \beta = \frac{C}{A}\). Calculating yields for sum: \(-\frac{6bc}{3b^2} = -\frac{2c}{b}\) and product: \(\frac{2c^2}{3b^2}\). The condition \(c^2 - 4ab < 0\) ensures the product \(\alpha \beta\) is positive since \(b^2\) cancels out any negative contributions.

Step 4: Determine Expression Behavior

Since \(c^2 < 4ab\), the quadratic form \(Ax^2 + Bx + C\) with positive leading coefficient \(3b^2\) and positive product \(\alpha\beta\) ensures the expression is always positive for all real \(x\). Furthermore, \(A(\alpha+\beta)^2 - 4AC\) simplifies to show positivity according to our condition.

Step 5: Compare with Given Values

Given the expression is positive and uses \(-c^2 < 4ab\), it must be greater than \(4ab\). Thus, \(3b^2x^2 + 6bc x + 2c^2 > 4ab\) satisfies the condition for all \(x\), matching option (A).

Key Concepts

Imaginary Roots

In the realm of quadratic equations, roots are solutions that solve the equation when set to zero. Sometimes, these roots can be imaginary. Imaginary roots occur when real roots do not satisfy the equation.
Imaginary numbers arise from the square root of a negative number. This comes into play when solving a quadratic equation. If the discriminant (the core part of the quadratic formula) is negative, the roots of the equation are not real numbers.For the quadratic equation of form \(ax^2 + bx + c = 0\), the discriminant is calculated as \(b^2 - 4ac\). When this value is less than zero, the equation's roots involve the imaginary unit \(i\), defined as \(i^2 = -1\). This means the equation has complex conjugate roots \(m + ni\) and \(m - ni\), where \(m\) and \(n\) are real numbers. In our exercise, since \(c^2 - 4ab < 0\), the equation \(bx^2 + cx + a = 0\) has imaginary roots.

Discriminant

The discriminant is a crucial element for understanding the nature of roots in a quadratic equation. It is given by \(\Delta = b^2 - 4ac\). The sign of the discriminant tells us about the type of roots we can expect.

• If \(\Delta > 0\), the quadratic equation has two distinct real roots.
• If \(\Delta = 0\), the quadratic equation has exactly one real root, or a repeated root.
• If \(\Delta < 0\), the quadratic equation has imaginary roots, meaning no real solutions are available.

In our specific problem, the discriminant \(c^2 - 4ab < 0\) indicates that the roots of the equation are imaginary. Recognizing the discriminant as negative helps us deduce certain properties of other expressions related to the equation, like its positivity in the context of real numbers.

Quadratic Expression Analysis

Analyzing a quadratic expression involves understanding the Sum and Product of its roots along with its behavior. In this case, we analyze the expression \(3b^2x^2 + 6bcx + 2c^2\). This expression forms another quadratic equation.
We use the relationships:

• Sum of roots: From a quadratic \(Ax^2 + Bx + C\), the sum is calculated as \(-\frac{B}{A}\).
• Product of roots: From the quadratic, this is calculated as \(\frac{C}{A}\).

The sum and product of roots provide insight into how the quadratic behaves graphically and numerically. Given our scenario from the exercise, we determine

• Sum of roots: \(-\frac{2c}{b}\)
• Product of roots: \(\frac{2c^2}{3b^2}\)

The equation \(c^2 - 4ab < 0\) ensures a positive product, meaning the quadratic opens upwards and is always positive. Therefore, this quadratic is greater than any choice value provided for all possible values of \(x\). Hence, we concluded that the quadratic is greater than the negative boundary set by the problem, consistent with option A.