(x-5).jpeg "(x-1)(x-5)")
Factoring and Solving (x-1)(x-5) = 0
The expression (x-1)(x-5) is a factored quadratic equation. Let's explore what this means and how to work with it.
Understanding the Factored Form
- Factoring: Factoring a quadratic equation means breaking it down into simpler expressions that multiply together to get the original expression. Here, (x-1) and (x-5) are the factors of the quadratic equation.
- Zero Product Property: The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero.
Solving for x
To solve for the values of x that satisfy the equation (x-1)(x-5) = 0, we can use the Zero Product Property:
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Set each factor equal to zero:
- x - 1 = 0
- x - 5 = 0
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Solve for x in each equation:
- x = 1
- x = 5
Therefore, the solutions to the equation (x-1)(x-5) = 0 are x = 1 and x = 5.
Expanding the Factored Form
To see how (x-1)(x-5) represents a quadratic equation, we can expand it using the distributive property:
(x-1)(x-5) = x(x-5) - 1(x-5) = x² - 5x - x + 5 = x² - 6x + 5
So, (x-1)(x-5) is equivalent to the quadratic equation x² - 6x + 5.
Applications of Factored Form
The factored form of a quadratic equation is particularly useful for:
- Finding the roots: The roots of a quadratic equation are the values of x that make the equation equal to zero. The factored form directly gives us the roots.
- Graphing: The roots of a quadratic equation correspond to the x-intercepts of its graph.
- Solving real-world problems: Many real-world situations can be modeled with quadratic equations, and factoring allows us to find solutions.
In summary, the factored form (x-1)(x-5) provides a straightforward way to solve the quadratic equation, find its roots, and understand its behavior.