Problem 111 - College Algebra

Question

Perform the indicated operations. $$ [(7 x+5)+4 y][(7 x+5)-4 y] $$

Step-by-Step Solution

Answer

The result of the operation \((7x+5+4y)(7x+5-4y)\) is \(49x^2+70x+25 - 16y^2\).

Step 1: Determine the format of the given problem

Check if the given expression follows the pattern of difference of squares \((a+b)(a-b)\). Here \(a\) is \(7x+5\) and \(b\) is \(4y\).

Step 2: Apply the formula of difference of squares

Replace \(a\) and \(b\) in the formula. This gives \((7x+5)^2 - (4y)^2\).

Step 3: Expand the squared terms

Solve \((7x+5)^2\) and \((4y)^2\). This result in \(49x^2+70x+25\) and \(16y^2\) respectively.

Step 4: Subtract the squared terms

Now subtract the second term from the first one to get the final result: \(49x^2+70x+25 - 16y^2\).

Key Concepts

Difference of Squares

One of the most useful identities in algebra is the difference of squares formula. This formula states that for any expressions \(a\) and \(b\), the product of the sum and difference, \((a+b)(a-b)\), can be expressed as \(a^2-b^2\).

By comparing the initial expression \([(7x+5)+4y][(7x+5)-4y]\) to this structure, you can see that \(a = 7x+5\) and \(b = 4y\). Recognizing this pattern allows us to simplify the problem significantly by directly applying the identity, transforming it into \((7x+5)^2 - (4y)^2\).

The key advantage here is that it reduces a complicated multiplication problem into simpler squares, making it much easier to handle.

Expanding Expressions

Expanding expressions is an essential skill in algebra, allowing you to simplify and better understand equations. Once we have identified the expression as a difference of squares, we need to expand \((7x+5)^2\) and \((4y)^2\).

Let's break it down: expanding \((7x+5)^2\) involves squaring a binomial, which requires the use of another known formula: \((a+b)^2 = a^2 + 2ab + b^2\). Applying this, we have:

• Calculate \( (7x)^2 = 49x^2 \)
• Calculate \( 2(7x)(5) = 70x \)
• Calculate \( 5^2 = 25 \)

So, \((7x+5)^2 = 49x^2 + 70x + 25\).

For \((4y)^2\), the process is straightforward: simply square \(4y\), resulting in \(16y^2\). Expanding both components helps to reveal the inner parts of the expression, making subtraction easier in subsequent steps.

Algebraic Operations

Once you've expanded the terms using algebraic identities, the final step is to perform algebraic operations to obtain the solution. The primary operation in this task is subtraction.

Following our expansion, we need to subtract the result of \((4y)^2\) from \((7x+5)^2\):

• The expression becomes \(49x^2 + 70x + 25 - 16y^2\).

By combining like and unlike terms appropriately, we simplify the problem to its final form. This step highlights the power of algebraic manipulation, as it transforms a potentially difficult problem into a simpler expression by systematically applying identities and operations.