## Understanding (a - 3b)^2

The expression (a - 3b)^2 represents the square of the binomial (a - 3b). This means we are multiplying the binomial by itself:

**(a - 3b)^2 = (a - 3b)(a - 3b)**

To expand this expression, we can use the **FOIL** method (First, Outer, Inner, Last):

**First:**Multiply the first terms of each binomial: a * a = a^2**Outer:**Multiply the outer terms: a * -3b = -3ab**Inner:**Multiply the inner terms: -3b * a = -3ab**Last:**Multiply the last terms: -3b * -3b = 9b^2

Now, combine the terms:

a^2 - 3ab - 3ab + 9b^2

Finally, simplify by combining the like terms:

**a^2 - 6ab + 9b^2**

Therefore, the expanded form of (a - 3b)^2 is **a^2 - 6ab + 9b^2**.

## Key Points

**Squaring a binomial:**Remember that squaring a binomial means multiplying it by itself.**FOIL method:**This is a helpful technique to expand binomials by systematically multiplying all the terms.**Combining like terms:**After applying FOIL, ensure to simplify the expression by combining terms with the same variables and exponents.

## Applications

Understanding how to expand binomials like (a - 3b)^2 is crucial in various mathematical concepts:

**Algebraic manipulation:**Expanding binomials is essential for simplifying expressions and solving equations.**Quadratic equations:**The expression (a - 3b)^2 can be a factor in quadratic equations, helping to solve for the roots.**Calculus:**Understanding binomial expansion is vital in topics like differentiation and integration.

By grasping the concept of expanding binomials, you can navigate more complex mathematical concepts with greater ease.