Algebraic Manipulation and Equation Solving
This page presents an algebraic exercise that covers development, factorization, and equation solving. The problem revolves around a quadratic expression and its various transformations.
The exercise begins with the expression A = ² - 64 and asks for three operations:
- Developing, reducing, and ordering A
- Factorizing A
- Solving the equation 3x - 10$$3x + 6 = 0
The solution process demonstrates several important algebraic techniques:
For the development part, the square of a binomial is expanded, and like terms are combined. This results in the standard form of a quadratic expression: 9x² - 12x - 60.
Example: A = ² - 64 = 9x² - 12x + 4 - 64 = 9x² - 12x - 60
In the factorization step, the expression is recognized as the difference of two squares, allowing for the application of the a² - b² identity.
Definition: The identity a² - b² = a - b$$a + b is a fundamental algebraic formula used for factorization.
The factorization process transforms the expression into 3x - 10$$3x + 6, which is then used in the equation-solving part.
Highlight: The factored form 3x - 10$$3x + 6 is crucial for solving the equation, as it allows for the application of the zero product property.
For solving the equation, the zero product property is applied, leading to two linear equations: 3x - 10 = 0 and 3x + 6 = 0.
Vocabulary: The zero product property states that if the product of factors is zero, then at least one of the factors must be zero.
These equations are solved to find the roots of the original quadratic expression:
x = 10/3 and x = -2
Quote: "Les solutions de l'équation sont : -2 et 10/3." (The solutions of the equation are: -2 and 10/3.)
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