Univariate quadratic equation ax? +bx+c=0？ (a≠0 and △=b? -4ac & gt； 0), let two roots be x 1 and x2 be X 1+X2=? -b/a，X 1 X2=c/a， 1/x 1+ 1/X2 =(x 1+X2)/x 1 X2 .

Judging the Root Quadratic Equation ax by Vieta Theorem? +bx+c=0 (a≠0), if b? -4ac & lt； 0 then the equation has no real root, if b? -4ac=0, then the equation has two equal real roots. If b? -4ac & gt； 0, the equation has two unequal real roots.

Theorem generalization: If two numbers are opposite to each other, then b=0. If two roots are reciprocal, then a = c, if one root is 0, then c=0. If one is-1, then? A-b+c=0. If one is 1, then? A+b+c=0. If A and C are different symbols, then the equation must have two real roots.

Vieta's theorem explains the relationship between roots and coefficients in a quadratic equation with one variable. Francois Viete, a French mathematician, established the relationship between the roots and coefficients of an equation in his book On the Identification and Correction of Equations, 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.

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.

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.

The discriminant of the root of a quadratic equation with one variable is △ = b? -4ac(a, B, C are quadratic term coefficients, linear term coefficients and constant terms of a quadratic equation). 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.

Vieta theorem can be used to quickly find out the relationship between the roots of two equations. Vieta theorem is widely used in elementary mathematics, analytic geometry, plane geometry and equation theory.