If \(\mathrm{b}^{2}=\) ac, equation \(\mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0\) and \(\mathrm{d} \mathrm{x}^{2}+2 \mathrm{ex}+\mathrm{f}=0\) have common roots then \((\mathrm{d} / \mathrm{a}),(\mathrm{e} / \mathrm{b}),(\mathrm{f} / \mathrm{c})\) are in (a) A.P. (b) G.P. (c) H.P. (d) None of these

Quadratic equations are an essential concept in algebra, central to understanding various aspects of mathematics, physics, engineering, and other scientific fields. Fundamentally, a quadratic equation is a polynomial equation of the second degree, generally represented as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients and \( a eq 0 \). The solutions to these equations, also known as the roots, can be real or complex numbers which satisfy the equation.

The roots can be found using various methods including factoring, completing the square, or using the quadratic formula \( x = (-b \pm \sqrt{b^2-4ac})/(2a) \). This formula provides a straightforward way to calculate roots, as long as one can efficiently handle the discriminant \( b^2-4ac \). If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root (a repeated root), and if negative, there are two complex roots.

Quadratic equations are not just about finding their roots; they also have profound implications in graphing a parabola, understanding projectile motion, and solving optimization problems. Their properties, such as their symmetry and vertex, give us a deeper understanding of quadratic functions and their applications.

A geometric progression (G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For instance, in the sequence \( 2, 4, 8, 16, ... \), the common ratio is 2 because each term is twice the preceding term.

A geometric progression can be finite or infinite and it is characterized by its first term \( a \) and the common ratio \( r \). The general form of a geometric sequence is \( a, ar, ar^2, ar^3, ... \). The nth term of a geometric progression can be expressed as \( ar^{(n-1)} \), and the sum of the first n terms in a finite G.P. is calculated using the formula: \[ S_n = a \frac{1-r^n}{1-r} \] assuming \( r eq 1 \).

Understanding geometric progressions is pivotal for solving problems pertaining to exponential growth and decay, population models, and financial mathematics such as compound interest calculations. The concept is also linked to the properties of quadratic equations when examining the relationship between their coefficients, as observed in the common roots problem.

The coefficients of a quadratic equation \( ax^2 + bx + c = 0 \) carry significant information about the equation's graph and its roots. Here's a quick breakdown of how each coefficient affects the quadratic equation:

• The leading coefficient \( a \) determines the direction of the parabolic curve (upward if positive, downward if negative) and affects the width of the parabola.
• The coefficient \( b \) influences the axis of symmetry of the parabola and partly determines the vertex's horizontal position.
• The constant term \( c \) represents the y-intercept of the parabola, which is where it crosses the y-axis.

The relationship between coefficients becomes particularly intriguing when examining the roots of quadratic equations with something called the 'conjugate root theorem' in the context of complex numbers, and the discriminant, which we mentioned earlier.

When two different quadratic equations share the same roots, their coefficients are bound in specific ways. In such cases, the relationship among the coefficients is not arbitrary but follows a pattern based on the roots' nature and sequence, be it arithmetic, geometric, or, less commonly, harmonic progressions. This concept is foundational when dealing with problems related to common roots as it lays the base for understanding the intricate connections between algebraic equations and their graphical representations.