Problem 30 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 4 \sqrt{3 x+3}=24 $$

Step-by-Step Solution

Answer

The solution for the equation is \(x = 11\).

Step 1: Isolate the square root

The first step is to isolate the square root. In our equation \(4\sqrt{3x+3} = 24\), we start by dividing both sides by 4 to get \(\sqrt{3x+3} = 6\).

Step 2: Remove the square root

We want to remove the square root. We can do that by squaring both sides of the equation: \((\sqrt{3x+3})^2=6^2\), which simplifies to \(3x+3=36\).

Step 3: Isolate \(x\)

Now, let's isolate the variable \(x\). First, subtract 3 from both sides to get \(3x = 33\), then divide both sides by 3 to isolate \(x\), so we get \(x = 11\).

Key Concepts

Square Roots

Square roots are fundamental to solving radical equations. They are incredibly useful in various fields, including algebra and geometry. A square root function is essentially the opposite of squaring a number. When you take the square root of a number, you're asking "What number, when multiplied by itself, gives me this number?"
For example, the square root of 25 is 5, because \(5 \times 5 = 25\).

• Square roots undo the operation of squaring a number.
• They appear in equations that need a specific approach to be solved.

Removing a square root from an equation requires careful attention, as it involves squaring both sides of the equation. This action modifies the equation to something more workable and often transforms it into a linear equation. Remember to always check your solutions, as squaring both sides can sometimes introduce extraneous solutions.

Isolating Variables

Isolating the variable is a critical step in solving equations. This process involves rearranging the equation term by term to get the variable alone on one side. In our example, we first isolated the square root by dividing by 4, which got us to \(\sqrt{3x+3} = 6\).

• This step simplifies the equation, making the variable clear.
• It’s crucial for identifying what operation to perform next to solve for the variable.

To isolate a variable effectively, use basic algebraic operations like addition, subtraction, multiplication, and division. Ensuring that each operation keeps the equation balanced is important, which means what you do to one side of the equation, you must do to the other. By isolating variables, you narrow down the search for the correct solution, creating a pathway towards finding the value of the unknown.

Algebraic Equations

Algebraic equations are foundational in mathematics and serve as precise, compact representations of relations between quantities. They consist of constants, variables, and arithmetic operations. Solving them involves finding values for the variables that make the equation true.

• They can be linear, quadratic, or involve higher-degree polynomials.
• Simplifying and solving algebraic equations entails a methodical approach.

In our example, we worked through a radical equation. Initially containing a square root, once isolated, the equation morphed into a simple linear equation: \(3x + 3 = 36\), which was then solved by standard algebraic techniques. These steps include arranging like terms and performing arithmetic operations to clear the variable of surrounding coefficients and constants. Understanding how to manipulate such equations is key to mastering a wide range of mathematical problems.