Problem 58 - Elementary Algebra

Question

The area of a square is given by the function \(A(x)=x_{2}+16 x+64,\) where \(x\) is measured in meters. Rewrite this function in factored form.

Step-by-Step Solution

Answer

The factored form of the function is \((x+8)^2\).

Step 1: Identify the Function

Given the function for the area, we have \( A(x) = x^2 + 16x + 64 \). Our goal is to rewrite this function in factored form.

Step 2: Find Two Numbers that Add to the Middle Coefficient

Look for two numbers that multiply to give the constant term \(64\) and add to give the middle coefficient \(16\). The numbers \(8\) and \(8\) fit this criterion, since \(8 \times 8 = 64\) and \(8 + 8 = 16\).

Step 3: Write as a Perfect Square Trinomial

Notice that the expression can be rewritten as a perfect square trinomial. The expression becomes \((x + 8)^2\), since \((x + 8)(x + 8) = x^2 + 16x + 64\).

Step 4: Verification

Verify by expanding \((x + 8)^2\) to check that it becomes \(x^2 + 16x + 64\). Expanding gives \(x^2 + 8x + 8x + 64 = x^2 + 16x + 64\), confirming the factorization.

Key Concepts

Understanding Perfect Square Trinomials

In algebra, a perfect square trinomial is a special type of expression that can be easily rewritten as the square of a binomial. A trinomial is a polynomial with three terms. Specifically, for it to be a perfect square trinomial, the expression must look like this:

• The square of a term (\(x^2\)
• A double product of two terms (\(2ab\)
• The square of a second term (\(b^2\)

This results in the format: \(a^2 + 2ab + b^2 = (a + b)^2\). This helps you see how the expression can be viewed as a *squared binomial*. For example, in the expression \(x^2 + 16x + 64\), the two numbers 8 and 8 satisfy:

• 8 multiplied by 8 gives us 64, which is our constant term
• 8 added to 8 gives us 16, which is the coefficient of the middle term

Thus, this expression can be rewritten as \((x + 8)^2\). This factored form reveals the underlying structure of the trinomial as a square, simplifying many algebraic operations and revealing its geometric interpretation.

Exploring Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication). They form the basis of algebra and are used to describe mathematical relationships. In the exercise, the expression \(A(x) = x^2 + 16x + 64\) is an algebraic expression representing the area of a square. Here,

• \(x^2\) stands for the area contribution from one side's variable length
• 16x represents a linear relationship where 16 times a variable length x adds to the area
• 64 is a constant area contribution

Algebraic expressions give us the opportunity to generalize mathematical calculations and offer a framework for algebraic manipulations such as factoring. By factoring such expressions, we uncover new properties or simplifications that make further calculations more manageable.

Relating to the Area of a Square

Understanding the area of a square is central to this problem. The area is the measure of space inside the square's boundary and is calculated as \( ext{side} imes ext{side} \). When expressed algebraically like in \(A(x) = x^2 + 16x + 64\), it provides insight into how different expressions relate to physical dimensions. In this case, the area can be viewed geometrically and algebraically through factoring:

• Geometrically, seeing \((x + 8)^2\) translates to each side of the square being \(x + 8\) meters long
• Algebraically, it shows the transformation of the expression into a form that indicates 'squaring' of the binomial

This perspective underscores a profound connection between algebraic manipulation and geometric interpretation, making abstract math tangible and enhancing understanding for students.