(x-7).jpeg "(x-2)(x-7)")
Factoring and Solving (x-2)(x-7)
The expression (x-2)(x-7) is a factored quadratic expression. This means it represents the product of two linear expressions: (x-2) and (x-7). Let's explore what this expression represents and how we can use it.
Understanding the Factored Form
The factored form of a quadratic expression gives us valuable information about its roots, also known as its x-intercepts. In this case:
- (x - 2) = 0 implies x = 2
- (x - 7) = 0 implies x = 7
These values, x = 2 and x = 7, are the roots of the quadratic equation represented by (x-2)(x-7).
Expanding the Expression
To understand the equation in its standard quadratic form, we can expand the factored expression:
(x - 2)(x - 7) = x² - 7x - 2x + 14 = x² - 9x + 14
This expanded form, x² - 9x + 14, represents the same quadratic equation as (x-2)(x-7).
Solving the Equation
To solve the equation (x-2)(x-7) = 0, we can use the fact that the product of two factors is zero if and only if one or both of the factors are zero. Therefore:
- x - 2 = 0 => x = 2
- x - 7 = 0 => x = 7
This confirms that the solutions to the equation are x = 2 and x = 7.
Applications
The factored form of a quadratic expression has many applications, including:
- Finding the x-intercepts of a parabola: The roots of the quadratic equation represent the x-intercepts of the parabola that the equation describes.
- Solving real-world problems: Quadratic equations often model real-world scenarios like projectile motion or area calculations. The factored form helps us find the solutions to these problems.
- Graphing parabolas: Knowing the roots of the quadratic equation allows us to plot the parabola's x-intercepts, which is a crucial step in sketching its graph.
In conclusion, understanding the factored form (x-2)(x-7) helps us grasp the roots of the quadratic equation, expand it into its standard form, and solve it to find its solutions. This knowledge is crucial for understanding quadratic equations and their applications in various fields.