Analyzing Quadratic Equation Roots and Coefficients

The question asks us to match different conditions on the coefficients \(a_1\) and \(a_2\) of a quadratic equation \(s^2 + a_1s + a_2 = 0\) with the nature of its roots. This is a fundamental concept in algebra, especially relevant in areas like control systems where the roots of characteristic equations determine system stability.

A general quadratic equation is given by \(as^2 + bs + c = 0\). The roots of this equation are given by the quadratic formula:

\(s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

The nature of the roots is determined by the value of the discriminant, \(\Delta = b^2 - 4ac\).

In the given equation \(s^2 + a_1s + a_2 = 0\), we have \(a=1\), \(b=a_1\), and \(c=a_2\). So, the discriminant is \(\Delta = a_1^2 - 4(1)(a_2) = a_1^2 - 4a_2\).

Let's analyze each condition provided in List I based on the discriminant and the quadratic formula:

Condition (A): \(a_1^2 > 4a_2\)

This condition means \(a_1^2 - 4a_2 > 0\). The discriminant \(\Delta > 0\).

When the discriminant is positive, the quadratic equation has two distinct real roots.

The roots are \(s = \frac{-a_1 \pm \sqrt{a_1^2 - 4a_2}}{2}\).

For the roots to be "Negative Real and Unequal" as mentioned in List II option (III), they must be real and distinct (which \(\Delta > 0\) guarantees) and both must be negative.

• The sum of the roots is \(\frac{-a_1 + \sqrt{a_1^2 - 4a_2}}{2} + \frac{-a_1 - \sqrt{a_1^2 - 4a_2}}{2} = \frac{-2a_1}{2} = -a_1\). For both roots to be negative, their sum must be negative. Thus, \(-a_1 < 0\), which means \(a_1 > 0\).
• The product of the roots is \(\left(\frac{-a_1 + \sqrt{a_1^2 - 4a_2}}{2}\right) \left(\frac{-a_1 - \sqrt{a_1^2 - 4a_2}}{2}\right) = \frac{(-a_1)^2 - (\sqrt{a_1^2 - 4a_2})^2}{4} = \frac{a_1^2 - (a_1^2 - 4a_2)}{4} = \frac{4a_2}{4} = a_2\). For both roots to be negative, their product must be positive. Thus, \(a_2 > 0\).

Assuming \(a_1 > 0\) and \(a_2 > 0\), the condition \(a_1^2 > 4a_2\) leads to two distinct negative real roots. This matches List II option (III).

Condition (B): \(a_1^2 = 4a_2\)

This condition means \(a_1^2 - 4a_2 = 0\). The discriminant \(\Delta = 0\).

When the discriminant is zero, the quadratic equation has exactly one real root (which is a repeated root).

The root is \(s = \frac{-a_1 \pm \sqrt{0}}{2} = \frac{-a_1}{2}\).

For the root to be "Negative real and equal" as mentioned in List II option (I), it must be real and negative. Since \(\Delta = 0\), the roots are equal and real. For the root \(\frac{-a_1}{2}\) to be negative, we need \(-a_1 < 0\), which means \(a_1 > 0\). If \(a_1 > 0\), then \(a_1^2 > 0\), and since \(a_1^2 = 4a_2\), it implies \(4a_2 > 0\), so \(a_2 > 0\).

Assuming \(a_1 > 0\), the condition \(a_1^2 = 4a_2\) leads to a negative real and equal root. This matches List II option (I).

Condition (C): \(a_1^2 < 4a_2\)

This condition means \(a_1^2 - 4a_2 < 0\). The discriminant \(\Delta < 0\).

When the discriminant is negative, the quadratic equation has two distinct complex conjugate roots.

The roots are \(s = \frac{-a_1 \pm \sqrt{a_1^2 - 4a_2}}{2} = \frac{-a_1 \pm \sqrt{-(4a_2 - a_1^2)}}{2} = \frac{-a_1 \pm i\sqrt{4a_2 - a_1^2}}{2}\).

These roots are in the form \(\alpha \pm i\beta\), where the real part is \(\alpha = -\frac{a_1}{2}\) and the imaginary part is \(\beta = \frac{\sqrt{4a_2 - a_1^2}}{2}\).

List II option (IV) mentions "Conjugate Complex (Real part negative)". The roots are indeed complex conjugates. For the real part \(\left(-\frac{a_1}{2}\right)\) to be negative, we need \(-a_1 < 0\), which implies \(a_1 > 0\). Also, since \(a_1^2 < 4a_2\) and \(a_1^2 \ge 0\), it must be that \(4a_2 > a_1^2 \ge 0\), which implies \(a_2 > 0\).

Assuming \(a_1 > 0\), the condition \(a_1^2 < 4a_2\) leads to conjugate complex roots with a negative real part. This matches List II option (IV).

Condition (D): \(a_1 = 0, a_2 \ne 0\)

Substitute \(a_1 = 0\) into the equation \(s^2 + a_1s + a_2 = 0\). This gives \(s^2 + 0 \cdot s + a_2 = 0\), which simplifies to \(s^2 + a_2 = 0\).

So, \(s^2 = -a_2\).

• If \(a_2 > 0\), then \(s^2 = -a_2\) implies \(s = \pm \sqrt{-a_2} = \pm i\sqrt{a_2}\). These are pure imaginary roots. Pure imaginary roots are a special case of conjugate complex roots where the real part is zero. List II option (II) is "Conjugate Imaginary". This fits the case when \(a_2 > 0\).
• If \(a_2 < 0\), then \(s^2 = -a_2\) implies \(s = \pm \sqrt{-a_2}\), where \(-a_2 > 0\). These are real roots, one positive and one negative (\(\pm \sqrt{|a_2|}\)). This scenario is not described in List II.

Given the options in List II, it is implied that for condition (D), the case where \(a_2 > 0\) leading to conjugate imaginary roots is intended. This matches List II option (II).

Summary of Matching Coefficients and Root Nature

Based on the analysis above, the matching is as follows:

This matching corresponds to: (A) - (III), (B) - (I), (C) - (IV), (D) - (II).

Revision Table: Quadratic Roots

Additional Information: Significance in Systems

In many engineering and physics applications, particularly in the analysis of linear time-invariant (LTI) systems (like electrical circuits, mechanical systems, control systems), the behavior and stability of the system are determined by the roots of a characteristic equation, which is often a polynomial equation like the quadratic one discussed here. The roots are also called poles of the system.

• If all roots have negative real parts, the system is typically stable.
• If any root has a positive real part, the system is typically unstable.
• If roots are pure imaginary (real part is zero), the system is typically marginally stable or oscillatory.
• Real roots correspond to exponential responses.
• Complex conjugate roots correspond to oscillatory responses.

The conditions on \(a_1\) and \(a_2\) in the question, especially the implied conditions \(a_1 > 0\) and \(a_2 > 0\) leading to negative real parts or pure imaginary roots, are relevant to stability analysis for second-order systems represented by the characteristic equation \(s^2 + a_1s + a_2 = 0\).

For a standard second-order system \(s^2 + 2\zeta\omega_n s + \omega_n^2 = 0\), where \(\zeta\) is the damping ratio and \(\omega_n\) is the natural frequency, we have \(a_1 = 2\zeta\omega_n\) and \(a_2 = \omega_n^2\).

• \(a_1 = 2\zeta\omega_n\), \(a_2 = \omega_n^2\)
• \(\Delta = a_1^2 - 4a_2 = (2\zeta\omega_n)^2 - 4\omega_n^2 = 4\zeta^2\omega_n^2 - 4\omega_n^2 = 4\omega_n^2(\zeta^2 - 1)\)
• If \(\zeta > 1\) (overdamped), \(\zeta^2 - 1 > 0\), so \(\Delta > 0\). Roots are real and unequal. If \(\omega_n > 0\) and \(\zeta > 0\), \(a_1 > 0, a_2 > 0\), roots are negative real. This matches (A).
• If \(\zeta = 1\) (critically damped), \(\zeta^2 - 1 = 0\), so \(\Delta = 0\). Root is real and equal. If \(\omega_n > 0\), \(a_1 = 2\omega_n > 0\), \(a_2 = \omega_n^2 > 0\), root is negative real. This matches (B).
• If \(0 < \zeta < 1\) (underdamped), \(\zeta^2 - 1 < 0\), so \(\Delta < 0\). Roots are complex conjugates. Real part is \(-\zeta\omega_n\). If \(\zeta > 0, \omega_n > 0\), real part is negative. This matches (C).
• If \(\zeta = 0\) (undamped), \(a_1 = 0\), \(a_2 = \omega_n^2 > 0\). Equation is \(s^2 + \omega_n^2 = 0\), roots are \(s = \pm i\omega_n\) (pure imaginary). This matches (D) when \(a_2 > 0\).