If the roots of the equation \(x^{2}-b x+c=0\) be two consecutive integers then \(b^{2}-4 c=\) (a) \(-2\) (b) \(-3\) (c) 3 (d) 1

Understanding the relationship between the roots of a quadratic equation and its coefficients is a fundamental aspect of algebra. This relationship is particularly embodied in the sum and product of roots. For the general quadratic equation in the standard form, \(ax^2 + bx + c = 0\), the sum of the roots \(\text{(\text{root}_1 + \text{root}_2)}\) can be found using \(-\frac{b}{a}\), and the product \(\text{(\text{root}_1 \times \text{root}_2)}\) is \(\frac{c}{a}\).

This stems from the fact that if \(\text{root}_1\) and \(\text{root}_2\) are the roots of the equation, then \(x^2 - (\text{root}_1 + \text{root}_2)x + \text{root}_1 \times \text{root}_2 = 0\) can be constructed, which simplifies to the above relationship when compared to the standard form. This concept is widely used to solve problems involving quadratic equations, especially when the roots have certain properties or relationships, such as being consecutive integers, as in the given textbook exercise.

A quadratic equation is an algebraic expression of the second degree, typically in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. The graph of a quadratic function is a parabola that opens upward if \(a > 0\) and downward if \(a < 0\).

The values of \(x\) that satisfy the quadratic equation are known as the roots. These can be real or complex numbers and are found through factoring, completing the square, or utilizing the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The expression under the square root, \(b^2 - 4ac\), is known as the discriminant and provides vital information about the nature of the roots. A positive discriminant indicates two distinct real roots, zero means one real root (repeated), and a negative discriminant suggests two complex roots.

Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. They are the building blocks of algebra and allow us to represent real-life quantities and their relationships abstractly. Expressions can be as simple as \(5x\) or as complex as \((x^2 + 3x + 2)/(x - 1)\).

The manipulation of these expressions according to algebraic principles is what enables us to solve equations. For our quadratic scenarios, expressing roots as consecutive integers leads us to set up systems of equations which when solved, help reveal more about the structure of the quadratic equation itself. Algebraic expressions can be factored, expanded, and simplified to uncover the value of the unknown variable or to compare against other equations for solutions.