Hệ thức Vi-ét và ứng dụng trong giải phương trình bậc hai

Viète's formulas, named after the 16th-century French mathematician François Viète, are a set of powerful relationships that connect the coefficients of a polynomial equation to the sums and products of its roots. These formulas are particularly useful in solving quadratic equations, which are polynomial equations of degree two. This article delves into the intricacies of Viète's formulas and explores their practical applications in solving quadratic equations.

Viète's formulas provide a direct link between the coefficients of a polynomial equation and the properties of its roots. For a quadratic equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0, the formulas state that the sum of the roots (x₁ + x₂) is equal to -b/a, and the product of the roots (x₁ * x₂) is equal to c/a. These relationships hold true regardless of the nature of the roots, whether they are real or complex.

<h2 style="font-weight: bold; margin: 12px 0;">Understanding Viète's Formulas</h2>

The derivation of Viète's formulas for quadratic equations is straightforward. Consider the quadratic equation ax² + bx + c = 0. Let x₁ and x₂ be the roots of this equation. We can factorize the equation as follows:

a(x - x₁)(x - x₂) = 0

Expanding the left-hand side, we get:

ax² - a(x₁ + x₂)x + ax₁x₂ = 0

Comparing the coefficients of this equation with the original quadratic equation, we obtain:

-a(x₁ + x₂) = b

ax₁x₂ = c

Solving for x₁ + x₂ and x₁ * x₂, we get:

x₁ + x₂ = -b/a

x₁ * x₂ = c/a

These are the Viète's formulas for quadratic equations.

<h2 style="font-weight: bold; margin: 12px 0;">Applications of Viète's Formulas in Solving Quadratic Equations</h2>

Viète's formulas offer a convenient and efficient method for solving quadratic equations. They provide a way to determine the roots of a quadratic equation without resorting to the quadratic formula. Here are some key applications:

* <strong style="font-weight: bold;">Finding the roots without using the quadratic formula:</strong> If the sum and product of the roots are known, Viète's formulas can be used to directly determine the roots. For example, if the sum of the roots is 5 and the product is 6, then the roots are 2 and 3.

* <strong style="font-weight: bold;">Verifying the roots:</strong> Once the roots of a quadratic equation are found, Viète's formulas can be used to verify their correctness. If the sum and product of the roots obtained using the quadratic formula satisfy Viète's formulas, then the roots are correct.

* <strong style="font-weight: bold;">Solving problems involving relationships between roots:</strong> Viète's formulas can be used to solve problems that involve relationships between the roots of a quadratic equation. For example, if the sum of the squares of the roots is known, Viète's formulas can be used to find the roots.

<h2 style="font-weight: bold; margin: 12px 0;">Conclusion</h2>

Viète's formulas are a fundamental tool in the study of quadratic equations. They provide a direct link between the coefficients of a quadratic equation and the properties of its roots. These formulas offer a convenient and efficient method for solving quadratic equations, verifying the roots, and solving problems involving relationships between roots. By understanding and applying Viète's formulas, we gain a deeper insight into the nature and behavior of quadratic equations.