Question

Решите по теореме Виета

Answer 1

Так, что значит[/tex]

Answer 2

Solving Equations using Vieta's Theorem

Vieta's theorem is a mathematical theorem that relates the coefficients of a polynomial equation to its roots. It provides a way to find the sum and product of the roots of a quadratic equation. The theorem states that for a quadratic equation of the form ax^2 + bx + c = 0, the sum of the roots is equal to -b/a and the product of the roots is equal to c/a.

Let's solve some equations using Vieta's theorem:

1. Example 1: x^2 + 3x + 2 = 0

In this equation, a = 1, b = 3, and c = 2. According to Vieta's theorem, the sum of the roots is -b/a and the product of the roots is c/a.

- Sum of the roots: -b/a = -3/1 = -3 - Product of the roots: c/a = 2/1 = 2

Therefore, the sum of the roots is -3 and the product of the roots is 2.

2. Example 2: 8x^2 + 3x - 18 = 0

In this equation, a = 8, b = 3, and c = -18. Applying Vieta's theorem:

- Sum of the roots: -b/a = -3/8 - Product of the roots: c/a = -18/8

Therefore, the sum of the roots is -3/8 and the product of the roots is -18/8.

3. Example 3: x^2 - 9x + 20 = 0

In this equation, a = 1, b = -9, and c = 20. Using Vieta's theorem:

- Sum of the roots: -b/a = 9/1 = 9 - Product of the roots: c/a = 20/1 = 20

Thus, the sum of the roots is 9 and the product of the roots is 20.

4. Example 4: x^2 - 5x + 6 = 0

In this equation, a = 1, b = -5, and c = 6. Applying Vieta's theorem:

- Sum of the roots: -b/a = 5/1 = 5 - Product of the roots: c/a = 6/1 = 6

Therefore, the sum of the roots is 5 and the product of the roots is 6.

These examples demonstrate how Vieta's theorem can be used to find the sum and product of the roots of a quadratic equation.