Problem 53 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ (2 x+y-3)(2 x+y+3) $$

Step-by-Step Solution

Answer

The simplified expression is \(4x^2 + 4xy + y^2 - 9\).

Step 1: Recognize the Form

Identify the expression as the product of two binomials, which can be simplified using the difference of squares formula. The expression is \((2x + y - 3)(2x + y + 3)\) and fits the form \((a-b)(a+b)\) where \(a = 2x + y\) and \(b = 3\).

Step 2: Apply Difference of Squares Formula

The difference of squares formula is \((a - b)(a + b) = a^2 - b^2\). Substitute \(a = 2x + y\) and \(b = 3\) into this formula to get the expression:\[(2x + y)^2 - 3^2\].

Step 3: Expand the Binomial Square

Expand the expression \((2x + y)^2\):\[(2x + y)^2 = (2x)^2 + 2(2x)(y) + y^2\].This simplifies to \[4x^2 + 4xy + y^2\].

Step 4: Simplify the Expression

Substitute the expanded form from Step 3 back into the formula from Step 2:\[4x^2 + 4xy + y^2 - 3^2\].Subtract \(3^2 = 9\):\[4x^2 + 4xy + y^2 - 9\].

Step 5: Finalize the Expression

Combine all terms to present the simplified expression:\[4x^2 + 4xy + y^2 - 9\]. This is the simplest form of the given operation.

Key Concepts

Binomial Expansion

A binomial is a polynomial that contains exactly two terms, and binomial expansion involves expanding expressions that are raised to a power or multiplied together. In the given exercise, we work with the product of two binomials: \((2x+y-3)(2x+y+3)\). When you multiply two binomials that have the form \((a-b)(a+b)\), you apply the difference of squares formula, which simplifies the multiplication process. Binomial expansion allows us to break down these expressions into simpler terms by using patterns and formulas.
Understanding which patterns, such as \((a\pm b)^2\), apply to a pair of binomials can greatly simplify the problem-solving process. In this case, we recognize the structure as a difference of squares, yielding the simple solution of \(a^2 - b^2\). The important part in binomial expansion is identifying the pattern, which leads us smoothly to simplification.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. In our exercise, \(2x + y - 3\) and \(2x + y + 3\) are both algebraic expressions. The practice of working with these expressions includes combining like terms, performing operations, and simplifying as much as possible.
The component expressions in the question aren't overly complex, but each term needs careful attention. Each expression can be broken down into terms that are added or subtracted, such as \(2x\), \(y\), and \(3\). Understanding how to manipulate these basics is crucial, as they are foundational to solving more advanced algebraic equations.

• To simplify, always look for common terms that can be combined.
• Understanding how each algebraic term functions in the equation can make the simplification process more intuitive.

Ultimately, the goal with algebraic expressions is to perform operations that maintain equality while reaching a simplified form.

Simplification of Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining their equality. In this exercise, the expression is \((2x+y-3)(2x+y+3)\), and the goal is to simplify it using the difference of squares formula.
Simplification is a critical step because it helps in reducing the complexity of expressions, making them easier to work with. We first recognize the binomial structure in order to apply the appropriate formula. Applying the formula \((a-b)(a+b) = a^2 - b^2\), we end up with \((2x+y)^2 - 3^2\).
Further simplification involves expanding \((2x+y)^2\) to obtain \[4x^2 + 4xy + y^2 - 9\].

• Each term in the expanded form is calculated carefully to make sure nothing gets missed. This reinforces the need for precision in simplification.
• By substituting back into the original equation, and performing final subtraction of \(3^2\), we attain a neatly simplified expression with distinct terms.

The key to simplification is frequent practice with different types of expressions, forming a strong base in algebraic manipulation techniques.