(x-10).jpeg "(x-4)(x-10)")
Factoring and Solving the Expression (x-4)(x-10)
The expression (x-4)(x-10) is a simple quadratic expression in factored form. Here's a breakdown of its properties, how to solve it, and its significance:
Understanding the Factored Form
The expression is already factored into two binomials: (x-4) and (x-10). This tells us that the expression represents the product of two linear terms.
Expanding the Expression
To understand the expression's standard quadratic form, we can expand it using the distributive property (FOIL method):
(x-4)(x-10) = x(x-10) - 4(x-10) = x² - 10x - 4x + 40 = x² - 14x + 40
Finding the Roots
The roots (or solutions) of an equation are the values of x that make the equation equal to zero. To find the roots of the equation (x-4)(x-10) = 0, we use the Zero Product Property:
- If the product of two or more factors is zero, then at least one of the factors must be zero.
Therefore, we set each factor equal to zero and solve:
- x - 4 = 0 => x = 4
- x - 10 = 0 => x = 10
This tells us that the expression (x-4)(x-10) equals zero when x = 4 or x = 10.
Significance of the Factored Form
The factored form (x-4)(x-10) provides several insights:
- Roots: The roots of the expression are directly evident as the values inside the parentheses.
- Graph: The factored form indicates that the graph of the quadratic equation y = (x-4)(x-10) intersects the x-axis at x = 4 and x = 10.
- Vertex: The factored form can help in finding the vertex of the parabola. The x-coordinate of the vertex is the midpoint of the roots (in this case, (4+10)/2 = 7).
Applications
The ability to factor quadratic expressions is crucial in many areas of mathematics, including:
- Solving quadratic equations: Factoring allows you to find the roots of a quadratic equation easily.
- Graphing quadratic functions: The factored form helps determine the x-intercepts of the graph.
- Algebraic manipulations: Factoring is a fundamental technique for simplifying expressions and solving equations.
In conclusion, understanding the factored form of (x-4)(x-10) provides insights into its roots, graphical representation, and applications in various mathematical contexts.