(x%2B1).jpeg "(x-2)(x+1)")
Factoring and Solving the Expression (x-2)(x+1)
The expression (x-2)(x+1) represents a factored quadratic equation. Let's break down its meaning and explore its applications.
Understanding the Factored Form
- Factoring: The expression is already in factored form. This means it's expressed as a product of two binomials: (x-2) and (x+1).
- Binomials: Each factor, (x-2) and (x+1), is a binomial, containing two terms.
Expanding the Expression
To understand the original quadratic equation, we can expand the factored form:
(x-2)(x+1) = x(x+1) - 2(x+1)
= x² + x - 2x - 2
= x² - x - 2
This tells us that (x-2)(x+1) is equivalent to the quadratic equation x² - x - 2.
Finding the Roots or Solutions
The factored form makes it easy to find the roots (or solutions) of the equation:
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Setting each factor to zero: To find the values of x where the expression equals zero, we set each factor equal to zero and solve:
- x - 2 = 0 => x = 2
- x + 1 = 0 => x = -1
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Roots: Therefore, the roots of the equation (x-2)(x+1) = 0 are x = 2 and x = -1.
Applications
Understanding factored forms like (x-2)(x+1) has various applications:
- Solving quadratic equations: Factoring allows us to easily find the solutions of a quadratic equation.
- Graphing parabolas: The roots of the equation represent the x-intercepts of the parabola represented by the quadratic.
- Analyzing relationships: Factored forms help us understand how different variables interact within a quadratic relationship.
In Conclusion
The expression (x-2)(x+1) is a simple yet powerful representation of a quadratic equation. By understanding its factored form, we can easily find its roots, expand it to its original form, and apply it to various mathematical and real-world problems.