(x-7).jpeg "(x-7)(x-7)")
Understanding (x-7)(x-7)
The expression (x-7)(x-7) represents the product of two identical binomial factors: (x-7) and (x-7). This type of multiplication is known as squaring a binomial.
Expanding the Expression
To expand this expression, we can use the FOIL method:
- First: x * x = x²
- Outer: x * -7 = -7x
- Inner: -7 * x = -7x
- Last: -7 * -7 = 49
Adding these terms together, we get:
(x-7)(x-7) = x² - 7x - 7x + 49
Simplifying the Result
Combining like terms, we arrive at the simplified form of the expression:
(x-7)(x-7) = x² - 14x + 49
Understanding the Pattern
Notice that the expanded form of (x-7)(x-7) follows a specific pattern:
- The first term is the square of the first term in the binomial (x²).
- The second term is twice the product of the two terms in the binomial (-14x).
- The third term is the square of the second term in the binomial (49).
This pattern applies to squaring any binomial, and can be represented by the general formula:
(a + b)² = a² + 2ab + b²
Applications
Understanding how to expand and simplify expressions like (x-7)(x-7) is essential in various mathematical contexts, including:
- Algebraic manipulations: Simplifying complex expressions, solving equations, and factoring polynomials.
- Calculus: Finding derivatives and integrals of functions.
- Geometry: Calculating areas and volumes of geometric shapes.
- Physics: Modeling physical phenomena and solving equations.
Conclusion
The expression (x-7)(x-7) illustrates a fundamental concept in algebra - squaring a binomial. By expanding and simplifying this expression, we gain a deeper understanding of algebraic manipulations and their applications in various fields.