(x-4).jpeg "(x-4)(x-4)")
Understanding (x-4)(x-4)
(x-4)(x-4) is a mathematical expression representing the multiplication of two identical binomials: (x-4) and (x-4). It's often written in a simplified form as (x-4)².
Expanding the Expression
To understand what (x-4)(x-4) represents, we need to expand it. This means multiplying each term in the first binomial by each term in the second binomial:
(x-4)(x-4) = x(x-4) - 4(x-4)
Now, we distribute the multiplication:
x(x-4) - 4(x-4) = x² - 4x - 4x + 16
Finally, we combine the like terms:
x² - 4x - 4x + 16 = x² - 8x + 16
The Result: A Quadratic Equation
The expanded form, x² - 8x + 16, is a quadratic equation. This means it's a polynomial with the highest power of x being 2.
Why it matters
Understanding how to expand expressions like (x-4)(x-4) is crucial for several reasons:
- Solving Equations: It allows us to simplify equations and solve for unknown variables.
- Graphing Functions: The expanded form helps us understand the shape and behavior of quadratic functions.
- Factorization: Recognizing patterns like (x-4)(x-4) helps us factorize complex expressions and simplify them.
Further Exploration
You can explore this concept further by:
- Trying different binomials: Change the numbers in the binomials and see how the resulting expression changes.
- Using a graphical calculator: Plot the graph of y = x² - 8x + 16 and observe its shape.
- Investigating the relationship between the original binomial and the expanded quadratic: Notice the connection between the coefficients of the terms and the original binomial.
By understanding the expansion and simplification of expressions like (x-4)(x-4), you gain a deeper understanding of quadratic equations and their applications.