MÉTODO DE PO SHEN LOH (EJEMPLO)

Introduction to Quadratic Equations Using the Pochenlo Method

Overview of the Pochenlo Method

• The instructor, Licenciado Bolívar, introduces the topic of solving quadratic equations using the Pochenlo method, which is based on axioms related to quadratic equations.

• Key properties are established: the sum of roots x_1 + x_2 = -B/A and their product x_1 cdot x_2 = C/A.

• The method requires transforming the equation so that the coefficient of x^2 equals one, allowing for easier identification of roots.

Identifying Elements in Quadratic Equations

• To apply this method effectively, one must identify coefficients from a standard quadratic equation format.

• The first result states that if we assume both roots are equal (i.e., x_1 = x_2), then they can be expressed as -B/2A.

Adjusting for Product Condition

• Acknowledging that while the sum condition holds true under this assumption, it does not satisfy the product condition.

• An additional value, denoted as mu, is introduced to adjust for this discrepancy in calculating roots.

Calculating Roots with Adjustments

Deriving Values

• The product condition leads to a difference of squares formula: left(-B/2A + muright)left(-B/2A - muright).

• Solving yields mu = pm sqrtB^2 - 4AC/4A^2, providing a means to find actual roots.

Final Root Calculation

• With values for both roots established as:

• x_1 = -B/2A + mu

• x_2 = -B/2A - mu

Examples Using the Pochenlo Method

First Example: Solving a Quadratic Equation

• The first example involves solving x^2 - x - 3 = 0. Coefficients identified are a = 1, b = -1, c = -3.

• From these coefficients:

• Sum of roots: 1 (-(-1)/1)

• Product of roots: -3 (c/a).

Assumptions and Adjustments

• Assuming both roots equal half their sum gives each root an initial value of 0.5. However, adjustments must be made since products do not match.

Finalizing Solutions

• After calculations involving adjustments with mu (mu=pmsqrt13/4), final solutions yield specific values for both roots.

Identifying Elements of a Quadratic Equation

Steps to Identify Coefficients and Roots

• The first step in identifying the elements of the quadratic equation is to divide by two, ensuring that the coefficient of the second-degree variable (x²) equals one. This leads to an expression where x₁ + x₂ equals -3/2.

• The coefficients are identified as follows: A = 2, B = -1, and C = -2. It is assumed that the roots will be equal to half of their sum, leading to x₁ and x₂ both being 3/4.

• Since the product does not equal -1, a value 'μ' is added and subtracted. Using the difference of squares results in μ equaling ±√(25/16), which simplifies to ±5/4.