91Ó°ÊÓ

A quadratic equation is a second-degree polynomial equation in a single variable, usually denoted as:\[ax^2 + bx + c = 0\]where:

• \(a\), \(b\), and \(c\) are constants with \(a eq 0\).
• The highest power of the variable is squared, making it a 'quadratic.'

The solutions to a quadratic equation are called its 'roots.' These can be found using methods like factoring, completing the square, or the quadratic formula. In the exercise, the roots were assumed to be consecutive integers, which helped us apply concepts like Vieta's formulas and the discriminant to solve for the value of \(b^2 - 4c\). Understanding the basic structure of quadratic equations is crucial because it lays the foundation for recognizing and employing other properties, like those discussed next. By knowing the role each term plays, we can manipulate the equation to learn more about its roots and graphs.

Vieta's formulas provide a direct relationship between the coefficients of a polynomial equation and its roots. For a quadratic equation:\[ax^2 + bx + c = 0\]if \(p\) and \(q\) are the roots, Vieta's formulas state that:

• The sum of the roots \(p + q = -\frac{b}{a}\).
• The product of the roots \(p \cdot q = \frac{c}{a}\).

In our problem, the quadratic equation was \(x^2 - bx + c = 0\) with the assumption of roots being \(n\) and \(n+1\). Using Vieta's formulas, we could set up equations:

• The sum \(n + (n+1) = b\), resulting in \(b = 2n + 1\).
• The product \(n(n + 1) = c\), leading to \(c = n^2 + n\).

By establishing these relationships, we simplified the problem significantly, allowing us to focus on the calculation of the discriminant to find the required value \(b^2 - 4c\). Vieta's formulas are a powerful tool for deducing unknown parameters in polynomial equations, enhancing our ability to solve them.

The discriminant is a key component in the quadratic formula and provides insight into the nature of the roots of a quadratic equation. Given a quadratic equation in the form:\[ax^2 + bx + c = 0\]the discriminant \(\Delta\) is calculated as:\[\Delta = b^2 - 4ac\]The value of the discriminant can tell us about the roots:

• If \(\Delta > 0\), there are two distinct real roots.
• If \(\Delta = 0\), there is one real root (the roots are identical).
• If \(\Delta < 0\), there are two complex roots.

In the given exercise, we needed to calculate \(b^2 - 4c\) with specific \(b\) and \(c\) values derived from the assumption of consecutive integer roots. This result was \[(2n + 1)^2 - 4(n^2 + n) = 1\]which implies the correct answer is option (d) 1. Understanding the role of the discriminant helps to predict the type of roots one might encounter, as well as assists in drawing conclusions about specific values in problem scenarios like these.