Solving Quadratic Equations and Understanding the Roots Using Vieta's Formulas

In the realm of algebra, quadratic equations play a crucial role in various mathematical and real-world applications. A fundamental concept in dealing with these equations is the use of Vieta's formulas, which provide a straightforward way to relate the coefficients of a polynomial to the sum and product of its roots. In this article, we will delve into a detailed exploration of how to solve the quadratic equation (x^2 x - 5 0) and understand the roots using Vieta's formulas.

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. It can be written in the standard form as:

(ax^2 bx c 0)

Where (a, b,) and (c) are constants, and (a eq 0).

Vietà's Formulas and Their Application

Vietà's formulas establish a relationship between the coefficients of a quadratic equation and the roots of that equation. For a quadratic equation (x^2 px q 0), the formulas are:

The sum of the roots, (alpha beta -frac{p}{1}) The product of the roots, (alpha cdot beta frac{q}{1})

Application to the Given Equation

Consider the quadratic equation:

(x^2 x - 5 0)

Here, (a 1), (p 1), and (c -5).

Step 1: Identifying the Roots

According to the quadratic formula, the roots of the equation are given by:

[ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

Plugging in the values, we get:

[ x frac{-1 pm sqrt{1^2 - 4 cdot 1 cdot (-5)}}{2 cdot 1} ]

This simplifies to:

[ x frac{-1 pm sqrt{21}}{2} ]

Thus, the roots are:

[ p frac{-1 sqrt{21}}{2} ]

[ q frac{-1 - sqrt{21}}{2} ]

Step 2: Verifying Using Vieta's Sum and Product Formulas

According to Vieta's formulas:

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

And for the product of the roots:

[ pq frac{-5}{1} -5 ]

Let's verify these using the roots we found:

[ p q frac{-1 sqrt{21}}{2} frac{-1 - sqrt{21}}{2} frac{-2}{2} -1 ]

[ pq left(frac{-1 sqrt{21}}{2}right) left(frac{-1 - sqrt{21}}{2}right) left(frac{-1 sqrt{21}}{2}right) left(frac{-1 - sqrt{21}}{2}right) ] [ frac{(1 - 21)}{4} frac{-20}{4} -5 ]

Step 3: General Expression for Roots

To further ensure the correctness, we can express the general form of the roots in a more detailed step:

From (2p^2 p - 10 0) and using the relationships established:

[ 2p^2 - p - 10 0 ]

The sum of roots, (p q -1), and the product, (pq -5), hold true.

Conclusion

Through a series of algebraic manipulations and the application of Vieta's formulas, we were able to determine the roots of the given quadratic equation accurately. This method not only provides a quick and efficient way to solve such problems but also underlines the importance of these fundamental relationships in algebra.

Key Takeaways

Vietà's formulas are powerful tools for solving quadratic equations. Sum and product of roots can be directly derived using the coefficients of the polynomial. Lenient practice with these formulas reinforces understanding and problem-solving skills.

Related Keywords

Vietà's Formulas, Quadratic Equations, Roots of Quadratic Equations