Problem 29 - College Algebra

Question

Solve cach equation in Exercises \(15-34\) by the square root property. $$(3 x+2)^{2}=9$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{1}{3}\) and \(x = -\frac{5}{3}\).

Step 1: Isolate the square term

We start with the original equation \((3x + 2)^2 = 9\). There's no need to change anything in this step as the square term is already isolated.

Step 2: Utilize square root property

Square root property states that if \(z^2 = a\), then \(z = \sqrt{a}\) or \(z = -\sqrt{a}\). We apply this to the equation and get \(3x + 2 = \sqrt{9}\) and \(3x + 2 = -\sqrt{9}\). This simplifies to \(3x + 2 = 3\) and \(3x + 2 = -3\).

Step 3: Solve for x

In each of the equations from the previous step, isolate \(x\) by subtracting 2 from both sides and then dividing by 3. For the first equation this gives \(x = \frac{1}{3}\), whereas for the second equation \(x = -\frac{5}{3}\).

Key Concepts

Solving Quadratic Equations

Quadratic equations appear frequently in algebra, and understanding how to solve them is crucial. A quadratic equation typically takes the form \(ax^2 + bx + c = 0\). However, in some cases, the equation can be expressed as a perfect square, like in our example: \((3x+2)^2 = 9\). This kind of equation can be solved using the square root property—a handy tool that helps to simplify the process. When dealing with an equation like this, our goal is to find the values of \(x\) that make the equation true. By recognizing this structure, we can efficiently apply mathematical properties to find solutions. Using the square root property effectively reduces the complexity by allowing us to work with simpler linear equations. Thus, solving quadratic equations often involves identifying patterns and applying appropriate mathematical techniques strategically.

Isolation of Terms

Isolation of terms is a critical step in solving equations, making it one of the primary techniques in algebra. In the context of quadratic equations, isolating the square term allows us to apply the square root property directly. This step involves manipulating the equation so that the squared term is on one side, and all other terms are moved to the opposite side. For the given equation \((3x+2)^2 = 9\), our job is simplified because the squared expression is already isolated. Had it not been, you might need to perform operations such as addition, subtraction, multiplication, or division to isolate the term. The goal is to reach a form where applying further mathematical operations like taking a square root becomes straightforward. Essentially, isolation prepares the equation for the application of the square root property or any other algebraic step necessary to reach the solution.

Simplifying Equations

Simplifying equations is an essential mathematical skill. It involves reducing equations to their simplest form to make them easier to solve or understand. After isolating the square term in equations like our example, you apply the square root property, simplifying expressions like \(\sqrt{9}\). Since the square root of 9 is both 3 and -3, equations \(3x + 2 = 3\) and \(3x + 2 = -3\) result. Simplifying further involves getting \(x\) on its own. Here, subtracting 2 from both sides simplifies each equation to \(3x = 1\) and \(3x = -5\). Lastly, divide by 3 to isolate \(x\), resulting in the final solutions \(x = \frac{1}{3}\) and \(x = -\frac{5}{3}\). Simplification isn't just a mathematical requirement but also a strategy to reduce complexity, making problem-solving more manageable.