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How to prove the theorem of Vieta

How to prove the theorem of Vieta

François Viète - the famous French mathematician.

Vieta theorem allows to solve quadratic equations by the simplified scheme, which as a result save time spent on the account.

But to better understand the essence of the theorem, it is necessary to gain insight into the formulation and prove it.

Theorem of Vieta

The essence of this technique is to findthe roots of quadratic equations without the help of the discriminant. For equations of the form x2 + bx + c = 0, where there are two different real roots, two statements is true.
The first statement says that the sum of the rootsThis equation is equivalent to the value of the coefficient of the variable x (in this case b), but with the opposite sign. Visually it looks like this: x1 + x2 = b?.
The second statement is not already associated with the sum, and withthe product of the same two roots. Equals is the product of the free factor, ie c. Or, x1 * x2 = c. Both these examples are solved in the system.
Vieta theorem greatly simplifies the decision, butIt has one limitation. Quadratic equation, the roots of which can be found using this technique, it should be given. In the above equation, the coefficient a, the one that stands in front of x2, is unity. Any equation can be reduced to such a view, dividing the first coefficient of the expression, but not always, this operation is rational.

The proof

For a start it should be recalled, as a traditionIt decided to seek the roots of a quadratic equation. The first and second discriminant through roots are, namely: x1 = (? -b- D) / 2, x2 = (-b + D?) / 2. Generally divided into 2a, but, as already mentioned, the theorem can be applied only when a = 1.
Vieta's Theorem it is known that the sum of the roots is equal to the second factor with a minus sign. This means that x1 + x2 = (-b-? D) / 2 + (-b +? D) / 2 =? 2b / 2 =? B.
The same is true for the works of unknownroots: x1 * x2 = (? -b- D) / 2 * (-b + D?) / 2 = (b2-D) / 4. In turn, D = b2-4c (again with a = 1). It turns out that the result is as follows: x1 * x2 = (b2- b2) / 4 + c = c.
From the simple proof we can draw only one conclusion: Vieta theorem is completely confirmed.

The second formulation and proof

Vieta's theorem has another interpretation. To be more precise, there is no interpretation, and formulation. The fact is that, if observed the same conditions as in the first case, there are two different real roots, then the theorem can be recorded by another formula.
This equation is as follows: x2 + bx + c = (x - x1) (x - x2). If the function P (x) intersect at two points x1 and x2, then it can be written as P (x) = (x - x1) (x - x2) * R (x). In the case where P has the second degree, namely the original look and expression, R is a prime number, namely 1. This is true, for the reason that otherwise the equality is not satisfied. x2 factor when disclosing the brackets must not be greater than one, and the expression must be a square.

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