(x-7).jpeg "(x-4)(x-7)")
Expanding and Understanding (x-4)(x-7)
In algebra, expressions like (x-4)(x-7) are often encountered. They represent the product of two binomials, which can be expanded to reveal a quadratic expression. Let's delve into the process of expanding this expression and explore its significance.
Expanding the Expression
To expand (x-4)(x-7), we employ the distributive property (often referred to as FOIL for "First, Outer, Inner, Last"). This method ensures we multiply each term in the first binomial by each term in the second binomial.
- First: x * x = x²
- Outer: x * -7 = -7x
- Inner: -4 * x = -4x
- Last: -4 * -7 = 28
Combining these terms, we get: x² - 7x - 4x + 28
Simplifying the expression by combining like terms: x² - 11x + 28
Understanding the Result
The expanded form, x² - 11x + 28, represents a quadratic expression. This means it is a polynomial with the highest power of x being 2. Quadratic expressions often describe parabolas when graphed.
Key Observations:
- The coefficient of the x² term: This is always 1 in our example, indicating the parabola will be symmetrical around the y-axis.
- The coefficient of the x term: -11 determines the location of the vertex of the parabola. A negative coefficient means the parabola opens downwards.
- The constant term: 28 represents the y-intercept of the parabola, where the graph crosses the y-axis.
Applications
Understanding the expansion of (x-4)(x-7) is fundamental in algebra and has various applications:
- Solving equations: The expanded form allows us to solve quadratic equations by setting the expression equal to zero.
- Factoring: Recognizing the original factored form helps us factor other quadratic expressions.
- Graphing parabolas: By understanding the coefficients, we can easily sketch the graph of the parabola representing the expression.
Conclusion
Expanding and understanding (x-4)(x-7) provides insight into the nature of quadratic expressions, their graphical representation, and their use in solving equations and factoring. This simple expression holds significant value in the realm of algebra and its applications.