Problem 4 - Elementary and Intermediate Algebra

Question

Fill in the blanks. Consider \(49 x^{2}-28 x y+4 y^{2}.\) The first term is the square of____. The last term is the square of____. The middle term is the opposite of twice the product of ____ and ____.

Step-by-Step Solution

Answer

The first term is the square of 7x. The last term is the square of 2y. The middle term is the opposite of twice the product of 7x and 2y.

Step 1: Identify the Square Roots of the First and Last Terms

First, we identify the square roots of the expressions that form perfect squares. The first term in the expression is \(49x^2\). The square root of \(49x^2\) is \(7x\) because \((7x)^2 = 49x^2\). The last term is \(4y^2\). The square root of \(4y^2\) is \(2y\) because \((2y)^2 = 4y^2\).

Step 2: Fill in the First Blank

The exercise asks for the first term to be expressed as the square of a specific expression. Based on our analysis, the first term \(49x^2\) is the square of \(7x\).

Step 3: Fill in the Second Blank

Similarly, the exercise asks for the last term to be expressed as the square of a specific expression. The last term \(4y^2\) is the square of \(2y\).

Step 4: Analyze the Middle Term

The middle term is \(-28xy\). According to the task, it should be the opposite of twice the product of the terms found in Steps 1 and 2. We compute \(2 \times 7x \times 2y = 28xy\). The opposite of this product is \(-28xy\), which matches the middle term in the original expression.

Step 5: Fill in the Third and Fourth Blanks

The question asks for the middle term to be the opposite of twice the product of two variables. Based on the analysis in Step 4, these variables are \(7x\) and \(2y\).

Key Concepts

Perfect Squares

In algebra, a perfect square is a number or expression that is the product of an expression multiplied by itself. Understanding perfect squares is crucial when dealing with certain algebraic expressions, especially when simplifying or factoring them.

Perfect squares help us to recognize patterns and make simplifications more straightforward. When dealing with expressions like \[49x^2 - 28xy + 4y^2\],
you can spot perfect squares by isolating the first and last terms. Let's break down the process:

• The first term is \(49x^2\), which is a perfect square because it can be expressed as \((7x)^2\).
• The last term, \(4y^2\), is also a perfect square, as it can be written as \((2y)^2\).

Recognizing these squares is the first step to factoring and simplifying the expression. The middle term, in this case, helps us understand the relationship between these perfect squares.

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions or numbers. For algebraic expressions like perfect squares, factoring turns out to be a useful tool.

Factoring is essential because it helps simplify expressions, solve equations, and understand the properties of algebraic expressions. For the expression \[49x^2 - 28xy + 4y^2\],
factoring involves using the squared terms we identified earlier:

• As recognized in the previous section, the expression consists of two perfect squares: \((7x)^2\) and \((2y)^2\).
• The middle term \(-28xy\) must be equivalent to \(-2(7x)(2y)\), which confirms the formula for the square of a binomial \((a - b)^2 = a^2 - 2ab + b^2\).

So, this expression factors neatly into \((7x - 2y)^2\). Factoring is essential not just for simplifying expressions but also for solving quadratic equations, as it reveals potential solutions if set equal to zero.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operators such as addition, subtraction, multiplication, and division. They are foundational in algebra and pre-calculus, representing numbers and relationships in an abstract form.

When working with expressions like \[49x^2 - 28xy + 4y^2\],
understanding the structure is crucial:

• Terms in an expression consist of products of numbers and variables. For example, \(49x^2\) and \(-28xy\) in the equation are separate terms.
• Coefficients like 49 and 4 are numbers that multiply the variables, indicating the expression's scale and structure.

Understanding how these components fit together enables you to manipulate and simplify expressions effectively. As you gain familiarity with algebraic expressions, you'll develop skills crucial for problem-solving in more complex mathematical contexts.