Problem 29 - Algebra and Trigonometry

Question

Solve equation by the square root property. $$ (3 x+2)^{2}=9 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x= \frac{\sqrt{9}-2}{3}\) and \(x= \frac{-\sqrt{9}-2}{3}\).

Step 1: Simplify the Equation

Rewrite the equation \((3x + 2)^{2} = 9\) to expand the term. So we have \[(3x+2)^{2}=(3x+2)(3x+2)=9\]

Step 2: Apply the Square Root Property

Once we isolate the squared term, apply the square root property to both sides of the equation leads to two potential solutions. The Square Root Property states if \(x^{2} = c\) then \(x = \sqrt{c}\) or \(x = -\sqrt{c}\).So we have \[3x + 2 = \sqrt{9}\] and \[3x + 2 = -\sqrt{9}\]

Step 3: Isolate x

Subtract 2 from both sides in order to solve for x in each solution. So the solutions for x are \[3x = \sqrt{9} - 2 \Rightarrow x = \frac{\sqrt{9}-2}{3}\] and \[3x = -\sqrt{9} - 2 \Rightarrow x = \frac{-\sqrt{9}-2}{3}\]

Key Concepts

Solving Equations

Solving equations involves finding the values of the unknowns that make the equation true. In the case of the given exercise, we want to find the value of \(x\) satisfying the equation \((3x + 2)^2 = 9\). The approach starts by simplifying and applying specific mathematical properties.

This typically includes identifying the equation type and iteratively manipulating the equation to isolate the unknown. Here, we use the Square Root Property, helping us to directly find potential solutions. Remember:

• First, ensure the equation is simplified.
• Identify operations that will help isolate the desired variable.
• Apply appropriate algebraic properties like taking square roots when dealing with squared terms.

Successfully solving equations often requires checking solutions to ensure they satisfy the original equation.

Algebraic Methods

Algebraic methods are the techniques or procedures used in algebra to solve equations and manipulate expressions. The exercise provided is solved using one popular algebraic method: the Square Root Property.

The basic idea is to take the square root on both sides of the equation after isolating the squared term. This reduces the complexity of the equation and gives two potential solutions due to the nature of square roots. When applying algebraic methods:

• Simplify expressions where possible.
• Carefully isolate the squared variable or expression before applying the square root.
• Remember there are often two solutions because \(\pm\sqrt{c}\) covers both the positive and negative roots.

These methods are fundamental for tackling quadratic equations and can be extended to more complex polynomials in advanced algebra.

Equation Simplification

Equation simplification is the process of reducing an equation to its simplest form to make it easier to handle. For the equation \((3x + 2)^2 = 9\), simplification involves expanding or transforming terms to reveal the underlying structure.

In this step-by-step solution, the first task was recognizing that \((3x + 2)^2\) is perfectly set for applying the square root property. However, simplifying doesn't always mean expanding; sometimes it means reformatting to make other operations straightforward. Here’s how simplification allows for the application of the Square Root Property:

• Avoid unnecessary expansions when the square form is straightforward for square rooting.
• Ensure both sides of the equation are simplified for easier manipulation.
• Check each transformational step to maintain equation balance.

Mastering equation simplification is key to efficient problem-solving in algebra and vital before applying complex algebraic rules or properties.