(x-2).jpeg "(x-8)(x-2)")
Factoring and Solving the Equation (x-8)(x-2)
The expression (x-8)(x-2) represents a factored quadratic equation. Here's a breakdown of its meaning, how to solve it, and its applications:
Understanding the Factored Form
- Factoring: The expression (x-8)(x-2) is already in factored form. This means it's written as the product of two binomials: (x-8) and (x-2).
- Binomials: Binomials are algebraic expressions with two terms.
- Product: The factored form represents the product of these two binomials.
Expanding the Expression
To see the original quadratic equation, we can expand the factored form using the distributive property (also known as FOIL):
(x - 8)(x - 2) = x(x - 2) - 8(x - 2) = x² - 2x - 8x + 16 = x² - 10x + 16
Solving for x
To find the values of x that make the equation true (i.e., make the expression equal to zero), we can use the Zero Product Property:
Zero Product Property: If the product of two or more factors is zero, then at least one of the factors must be zero.
Applying this to our factored form:
(x - 8)(x - 2) = 0
This means either:
- x - 8 = 0 => x = 8
- x - 2 = 0 => x = 2
Therefore, the solutions to the equation (x - 8)(x - 2) = 0 are x = 8 and x = 2.
Applications
Factoring quadratic expressions is a fundamental skill in algebra and has applications in various areas, including:
- Solving Quadratic Equations: As demonstrated above, factoring helps find the solutions to quadratic equations.
- Graphing Quadratic Functions: The factored form reveals the x-intercepts of the parabola represented by the quadratic function.
- Optimization Problems: Finding the maximum or minimum values of quadratic expressions often involves factoring.
- Real-World Problems: Quadratic equations model many real-world situations, from projectile motion to finding the area of a rectangle.
Conclusion
The expression (x - 8)(x - 2) is a factored quadratic equation. By understanding the Zero Product Property, we can solve for the values of x that make the equation true. Factoring is a crucial concept in algebra with numerous applications in mathematics and various real-world scenarios.