## 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.