What is Vieta's formulas? Which grade's curriculum in Vietnam includes Vieta's formulas?

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What is Vieta's formulas? Which grade's curriculum in Vietnam includes Vieta's formulas?

What is Vieta's formulas?

Vieta's formulas: The relationship between roots and coefficients of a quadratic equation

Vieta's formulas is an important tool in algebra, helping us explore the relationship between the roots of a quadratic equation and its coefficients.

*The Significance of Vieta's formulas

Quick equation solving: If we know the sum and product of two roots, we can form the quadratic equation without solving the original equation.

Verification of roots: After finding the roots, we can use Vieta's formulas to verify the results.

Solving problems related to the roots of a quadratic equation: Vieta's formulas helps us tackle more complex problems, such as finding the maximum or minimum values of expressions containing the roots.

Content of the formulas

For the quadratic equation: ax^2 + bx + c = 0 (with a ≠ 0) that has two roots x1 and x2. In this case, we have:

Sum of the two roots: x1 + x2 = -b/a

Product of the two roots: x1 * x2 = c/a

*Example

Given the equation x^2 - 5x + 6 = 0. Without solving the equation, calculate:

Sum of the two roots

Product of the two roots

Solution:

Applying Vieta's formulas, we have:

Sum of the two roots: x1 + x2 = -(-5)/1 = 5

Product of the two roots: x1 * x2 = 6/1 = 6

*Note

Vieta's formulas only applies to quadratic equations that have two roots.

When applying Vieta's formulas, be aware of the signs of the coefficients a, b, c.

Example 1: Finding two numbers given their sum and product

Problem: Find two numbers given that their sum is 5 and their product is 6.

Solution: Let the two numbers be x and y. According to the problem, we have the system of equations:

x + y = 5

x * y = 6

This corresponds to the Viète's relationships for the quadratic equation with roots x and y. Thus, the quadratic equation is: x^2 - 5x + 6 = 0 Solving this equation, we get x = 2 and x = 3. Hence, the two numbers are 2 and 3.

Example 2: Verifying the roots of an equation

Problem: Verify whether x = 2 and x = -3 are the roots of the equation x^2 + x - 6 = 0.

Solution: Applying Vieta's formulas:

Theoretical sum of the two roots: -b/a = -1/1 = -1

Theoretical product of the two roots: c/a = -6/1 = -6

Given sum of the two roots: 2 + (-3) = -1

Given product of the two roots: 2 * (-3) = -6

We see that the given sum and product of the two roots match the theoretical values. Thus, x = 2 and x = -3 are the roots of the equation.

Example 3: Finding m such that the equation has two roots satisfying certain conditions

Problem: Find m such that the equation x^2 - 2mx + m^2 - 1 = 0 has two distinct roots x1, x2 satisfying x1^2 + x2^2 = 10.

Solution: For the equation to have two distinct roots, Δ > 0 ⇔ (-2m)^2 - 4(m^2 - 1) > 0 ⇔ 4 > 0 (always true) Applying Vieta's formulas:

x1 + x2 = 2m

x1 * x2 = m^2 - 1

We have: x1^2 + x2^2 = (x1 + x2)^2 - 2x1*x2 = (2m)^2 - 2(m^2 - 1) = 10 Solving this equation, we find m.

Example 4: Formulating a quadratic equation given the sum and product of roots

Problem: Formulate a quadratic equation with roots 2 and -3.

Solution: According to Vieta's formulas, we have:

Sum of the two roots: S = 2 + (-3) = -1

Product of the two roots: P = 2 * (-3) = -6

The desired quadratic equation is: x^2 - Sx + P = 0 ⇔ x^2 + x - 6 = 0

What is Viète's Theorem? At what grade level is Viète's Theorem introduced in the curriculum?

What is Vieta's formulas? Which grade's curriculum in Vietnam includes Vieta's formulas? (Image from the Internet)

Which grade's curriculum in Vietnam includes Vieta's formulas?

According to the Mathematics curriculum issued in conjunction with Circular 32/2018/TT-BGDDT, Vieta's formulas, fully known as Viète,

Based on Section V of the Appendix for the Mathematics curriculum issued in conjunction with Circular 32/2018/TT-BGDDT, the requirements for writing skills in Grade 9 Mathematics are as follows:

In Grade 9, as part of the section on Equations and Systems of Equations, students will learn about Quadratic Equations in One Variable. Vieta's formulas includes the following student requirements:

- Identify the concept of a quadratic equation in one variable. Solve quadratic equations in one variable.

- Calculate the root of a quadratic equation in one variable using a calculator.

- Explain Vieta's formulas and its applications (e.g., mental calculation of the roots of a quadratic equation, finding two numbers given their sum and product, etc.).

- Apply quadratic equations to solve practical problems.

Thus, according to regulations, students learn Vieta's formulas in Grade 9.

What happens if lower secondary school students in Vietnam do not take a test required for teacher evaluation?

Based on the regulations in Article 7 of Circular 22/2021/TT-BGDDT regarding periodic evaluation for lower secondary school students, the evaluation methods are as follows:

- Periodic evaluation (not applicable to groups of specialized study topics), includes midterm and end-of-term evaluations, conducted through: tests (on paper or computer), practical exercises, or study projects.

+ The duration for tests (on paper or computer) for a subject (excluding groups of specialized study topics) with up to 70 periods per school year is 45 minutes; for subjects (excluding groups of specialized study topics) with more than 70 periods per school year, the duration is between 60 and 90 minutes; for specialized subjects, a maximum of 120 minutes.

+ For tests (on paper or computer) evaluated by score, the test is designed based on a matrix, a description of the test, and must meet the requirements as specified in the Educational Program.

+ For tests (on paper or computer) evaluated by comments, practical exercises, or study projects, there must be guidelines and evaluation criteria according to the subject requirements as specified in the Educational Program before conducting.

- Each semester, every subject evaluated by comments has 01 (one) midterm evaluation and 01 (one) end-of-term evaluation.

- Each semester, every subject evaluated by comments combined with scores has 01 (one) midterm score (abbreviated as DDGgk) and 01 (one) end-of-term score (abbreviated as DDGck).

- Students who do not participate in the required number of tests and evaluations due to unavoidable reasons are allowed to make up the outstanding evaluations, with requirements equivalent to the missed tests. The make-up tests are conducted each semester.

- If a student does not participate in the make-up tests, they will be evaluated as Not Achieved or given a score of 0 (zero) for the missing evaluation.

Thus, lower secondary school students who do not take a required test for teacher evaluation will be given a make-up test. If the student does not participate in the make-up test, they will be evaluated as Not Achieved or given a score of 0 (zero).

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