Understanding the Sum and Product of Roots in Quadratic Equations

When dealing with quadratic equations, it is essential to understand the relationship between the coefficients of the equation and the roots. For a quadratic equation of the form:

[ax^2 - bx c 0]

we can derive some fundamental properties related to the roots, denoted as (p) and (q).

The Sum of the Roots

According to Vieta's formulas, the sum of the roots can be determined using the coefficients of the quadratic equation. Specifically, the sum of the roots (p) and (q) is given by:

[p q -frac{text{coefficient of } x}{text{coefficient of } x^2} -frac{-b}{a} frac{b}{a}]

This means that the sum of the roots of any quadratic equation (ax^2 - bx c 0) is:

[p q frac{b}{a}]

By simplifying, we see that the sum of the roots is simply:

[p q boxed{frac{b}{a}}]

The Product of the Roots

Additionally, Vieta's formulas also provide the product of the roots. For the same quadratic equation:

[ax^2 - bx c 0]

the product of the roots (p) and (q) is given by:

[p cdot q frac{text{constant term}}{text{coefficient of } x^2} frac{c}{a}]

Example and Application

Let's consider a specific quadratic equation to solidify our understanding. If we have the quadratic equation:

[ax^2 - bx c 0]

with specific values for (a), (b), and (c), we can apply the formulas to find the sum and the product of the roots:

Sum of roots (p q frac{b}{a}) Product of roots (p cdot q frac{c}{a})

For instance, if (a 1), (b 5), and (c 6), the quadratic equation becomes:

[x^2 - 5x 6 0]

Here, the sum of the roots (p q) is:

[p q frac{5}{1} 5]

And the product of the roots (p cdot q) is:

[p cdot q frac{6}{1} 6]

Conclusion

Understanding the sum and product of the roots of a quadratic equation is crucial in various mathematical applications, including solving equations, graphing, and analyzing the behavior of functions. By using Vieta's formulas, we can derive these values directly from the coefficients of the quadratic equation, simplifying many problem-solving processes.

In summary, for a quadratic equation (ax^2 - bx c 0):

[boxed{p q frac{b}{a}}]

[boxed{p cdot q frac{c}{a}}]