Problem 52 - College Algebra

Question

A statue is to be placed in the center of the park. The area of the base of the statue is \(4 x^{2}+12 x+9 \mathrm{~m}^{2}\). Factor the area to fi \(\mathrm{d}\) the lengths of the sides of the statue.

Step-by-Step Solution

Answer

The side lengths of the statue's base are \((2x + 3)\) meters each.

Step 1: Identify the Formula

The problem provides you with the expression for the area of the base of the statue: \(4x^2 + 12x + 9\). We need to factor this expression to find the lengths of the sides of the statue.

Step 2: Recognize the Quadratic Form

You need to approach this problem by factoring the quadratic expression, which can be rewritten in the general form \(ax^2 + bx + c\). In this case, \(a = 4\), \(b = 12\), and \(c = 9\).

Step 3: Find the Perfect Square

Notice that \(4x^2 + 12x + 9\) appears to be a perfect square trinomial. When a quadratic is a perfect square trinomial, it can be factored as \((ax + b)^2\).

Step 4: Confirm and Factorize

Check if \((2x + 3)^2 = 4x^2 + 12x + 9\).Calculate: \((2x + 3)(2x + 3) = 4x^2 + 6x + 6x + 9 = 4x^2 + 12x + 9\).Since they match, the factorization is correct.

Step 5: Write the Factorization

The area of the base of the statue \(4x^2 + 12x + 9\) can be factored into \((2x + 3)^2\). Thus, the lengths of the sides of the statue base are \(2x + 3\) meters each.

Key Concepts

Quadratic Expressions

A quadratic expression is a polynomial of degree two, typically in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). Quadratic expressions frequently appear in various real-world applications, such as calculating areas, modeling natural phenomena, and solving geometric problems. Understanding how to manipulate and factor these expressions is crucial in unleashing numerous mathematical solutions.
To factor a quadratic expression, the main goal is to rewrite it as a product of simpler expressions. Finding the correct factors requires identifying patterns and applying specific algebraic techniques. Factoring helps simplify problems, making them more manageable for solving equations or calculating values like lengths or areas. Focusing on what's given in your problem, such as coefficients \(a\), \(b\), and \(c\), guides you in selecting the right strategy to tackle quadratic expressions.

Perfect Square Trinomials

Perfect square trinomials are a special type of quadratic expression where the expression can be written as the square of a binomial. In other words, a trinomial is a perfect square if it takes the form \( (ax + b)^2\). Recognizing these patterns simplifies the factoring process, saving time and effort.
To identify a perfect square trinomial, you should notice if:

• Both the first and last terms are perfect squares.
• The middle term is twice the product of the square roots of the first and last terms.

In the given exercise, the expression \(4x^2 + 12x + 9\) was noticed to be a perfect square trinomial. This recognition on Step 4 allowed the solution to rewrite the expression as \((2x + 3)^2\), confirming that its factorization is straightforward. This factorization not only assists in solving the problem but reinforces the importance of understanding perfect square trinomials within algebraic solutions.

Algebraic Techniques

Algebraic techniques involve a range of methods used to manipulate expressions and solve equations. When dealing with quadratic expressions, these techniques are vital in breaking down complex problems into simple, manageable tasks. Strategies like recognizing standard forms, applying the distributive property, and confirming factorization are central to solving quadratic problems.
In our specific case, algebraic techniques helped identify that \(4x^2 + 12x + 9\) was not just any quadratic, but a perfect square trinomial. Understanding multiplication and distribution confirmed this, as demonstrated in Step 4 where \((2x + 3)(2x + 3)\) was expanded to become \(4x^2 + 12x + 9\). By confirming these patterns, students can more easily solve problems and make meaningful connections between different algebraic expressions.
Applying these techniques consistently enhances problem-solving skills, enabling students to tackle not only textbook exercises but also real-life scenarios requiring algebraic solutions. Cultivating a strong foundation in algebraic techniques empowers learners to engage with mathematics more confidently and creatively.