Analysis of Vieta Theorem
Francois Viete, a French mathematician, established the relationship between the roots and coefficients of equations in his book "Identification and Revision of Equations in 16 15" and put forward this theorem. Because David first discovered this relationship between the roots and coefficients of algebraic equations, people call this relationship the Vieta theorem.
Vieta theorem relation
Let the unary quadratic equation ax+bx+c=0(a, b, c∈R, a≠0), and the two x 1 and x2 have the following relationship:
x+x=-a/b xx=a/c
Generalization of Vieta Theorem
Inverse Theorem If two numbers α and β satisfy the following relations: α+β=-a/b and α β = a/c, then these two numbers α and β are the roots of the equation ax+bx+c=0(a, B, c∈R, a≠0).
Through the inverse theorem of Vieta's theorem, we can use the sum-product relation of two numbers to construct a quadratic equation with one variable.
A Brief History of the Development of Vieta Theorem
16 15 years, French mathematician Francois Viete improved the solutions of cubic and quartic equations in his book "On the Identification and Correction of Equations", and also established the relationship between the roots and coefficients of equations when n=2 and 3, which is the modern Vieta theorem.
David first discovered this relationship between the roots and coefficients of algebraic equations, so people called this relationship Vieta Theorem. David got this theorem in16th century. The proof of this theorem depends on the basic theorem of algebra, but the basic theorem of algebra was first discussed by Gauss in 1799.
The Significance of Vieta Theorem
Vieta's theorem plays a unique role in finding the symmetric function of roots, discussing the sign of roots of quadratic equations, solving symmetric equations and solving some conic problems. =b-4ac
The discriminant of the root of a quadratic equation with one variable is (A, B and C are quadratic coefficient, linear coefficient and constant term of the quadratic equation with one variable respectively). The relationship between Vieta's theorem and the discriminant of roots is even more inseparable.
The discriminant of roots is a necessary and sufficient condition for judging whether an equation has real roots. Vieta theorem explains the relationship between roots and coefficients. Whether the equation has real roots or not, Vieta's theorem is applicable between the roots and coefficients of a quadratic equation with real coefficients. The combination of discriminant and Vieta's theorem can more effectively explain and judge the conditions and characteristics of the roots of a quadratic equation with one variable.
The most important contribution of Vieta's theorem is the promotion of algebra. Firstly, he systematically introduced algebraic symbols, promoted the development of equation theory, replaced unknowns with letters, and pointed out the relationship between roots and coefficients. Vieta's theorem laid a foundation for the study of the unary equation in mathematics, and created and opened up a broad development space for the application of the unary equation.