(a-11).jpeg "(a-1)(a-11)")
Exploring the Expression (a-1)(a-11)
The expression (a-1)(a-11) represents the product of two binomials. Let's delve into its different aspects and applications.
Expanding the Expression
To fully understand the expression, we can expand it using the FOIL method (First, Outer, Inner, Last):
- First: (a * a) = a²
- Outer: (a * -11) = -11a
- Inner: (-1 * a) = -a
- Last: (-1 * -11) = 11
Combining the terms, we get: ** (a-1)(a-11) = a² - 11a - a + 11**
Simplifying further: ** (a-1)(a-11) = a² - 12a + 11**
Factoring and Finding Roots
The expanded form (a² - 12a + 11) reveals that the expression is a quadratic equation. We can factor it back to its original form:
- Find two numbers that add up to -12 and multiply to 11. These numbers are -1 and -11.
- Rewrite the expression using these numbers: (a - 1)(a - 11)
To find the roots (values of 'a' where the expression equals zero), we set each factor equal to zero:
- a - 1 = 0 => a = 1
- a - 11 = 0 => a = 11
Therefore, the roots of the equation (a-1)(a-11) are a = 1 and a = 11.
Applications
This expression can be useful in various contexts, including:
- Solving quadratic equations: Understanding the factored form allows us to easily solve equations like (a-1)(a-11) = 0.
- Graphing quadratic functions: The roots of the expression represent the x-intercepts of the parabola defined by the function y = (a-1)(a-11).
- Algebraic manipulations: The expression can be used in various algebraic manipulations and simplifications.
Conclusion
Understanding the expression (a-1)(a-11) is crucial for working with quadratic equations and functions. By expanding, factoring, and finding its roots, we gain valuable insights into its mathematical properties and applications.