Expanding and Simplifying Algebraic Expressions: 5(3a+b)² + 6(3a+b) - 8
In algebra, we often encounter expressions that involve variables, constants, and various mathematical operations. One important aspect of algebra is expanding and simplifying expressions, which can help us solve equations and inequalities more efficiently. In this article, we will explore the expansion and simplification of the expression 5(3a+b)² + 6(3a+b) - 8.
Step 1: Expand the Squared Expression
To begin, let's focus on the squared expression 5(3a+b)². To expand this expression, we need to use the formula for the square of a binomial:
(a + b)² = a² + 2ab + b²
Using this formula, we can expand 5(3a+b)² as follows:
5(3a+b)² = 5((3a)² + 2(3a)b + b²) = 5(9a² + 6ab + b²) = 45a² + 30ab + 5b²
Step 2: Expand the Remaining Expression
Next, we need to expand the expression 6(3a+b). This is a simple multiplication:
6(3a+b) = 18a + 6b
Step 3: Combine the Expanded Expressions
Now, let's combine the expanded expressions:
5(3a+b)² + 6(3a+b) - 8 = 45a² + 30ab + 5b² + 18a + 6b - 8
Step 4: Simplify the Expression
Finally, we can simplify the expression by combining like terms:
45a² + 30ab + 5b² + 18a + 6b - 8 = 45a² + 30ab + 5b² + 18a + 6b - 8 = 45a² + 30ab + 5b² + 18a + 6b - 8 = 45a² + 30ab + 5b² + 18a + 6b - 8
After simplifying, we are left with the final expression:
45a² + 30ab + 5b² + 18a + 6b - 8
This is the simplified form of the original expression 5(3a+b)² + 6(3a+b) - 8.
