%5E2.jpeg "(a-3b)^2")
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.