Problem 13 - Elementary Algebra

Question

For the following problems, solve the square root equations. $$ \sqrt{x+3}=3 $$

Step-by-Step Solution

Answer

Answer: x = 6

Step 1: Write down the given equation

We are given the equation: $$ \sqrt{x+3}=3 $$

Step 2: Square both sides of the equation

To remove the square root, we will square both sides of the equation: $$ (\sqrt{x+3})^2=3^2 $$

Step 3: Simplify the equation

Squaring both sides, we get: $$ x + 3 = 9 $$

Step 4: Solve for x

Now we will subtract 3 from both sides of the equation to isolate x: $$ x = 9 - 3 $$ $$ x = 6 $$ The solution is x = 6.

Key Concepts

Solving Square Root Equations

Solving equations that contain a square root, such as \( \sqrt{x+3} = 3 \), involves eliminating the square root to simplify and eventually isolate the variable. The primary technique for getting rid of a square root is to square both sides of the equation. This method is effective since the square of a square root returns the original value under the root. Here’s a simplified breakdown:- Start with the given equation: \( \sqrt{x+3} = 3 \).- Square both sides to eliminate the square root: \( (\sqrt{x+3})^2 = 3^2 \).
After squaring, you’ll have \( x+3 = 9 \). Now, the equation is simpler and free of square roots, allowing you to isolate the variable.
By adhering to this process, you can solve more complex equations involving square roots efficiently, expanding your problem-solving abilities in mathematics.

Understanding Algebra Through Equations

Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. It serves as a crucial tool for solving equations like \( \sqrt{x+3} = 3 \). Let's dive into how algebra helps in these situations:- **Variables and Constants:** In our equation, \( x \) is the variable, representing an unknown, while 3 is a constant, representing a known value.- **Balancing Equations:** The core principle of solving equations is maintaining balance. Whatever you do to one side, you must do to the other. Squaring both sides of \( \sqrt{x+3} = 3 \) shows this balancing act: \((\sqrt{x+3})^2 = 3^2\).- **Isolating the Variable:** After simplifying to \( x+3 = 9 \), you perform operations to isolate \( x \): subtract 3 from both sides, resulting in \( x = 6 \).
Through algebra, we not only find values for variables but also explore underlying mathematical relationships. This understanding is foundational for higher-level math concepts.

The Importance of Mathematics Education

Engaging in exercises like solving square root equations is an essential part of mathematics education. These types of problems help in developing logical and analytical skills. - **Critical Thinking:** When students solve equations, they learn to think critically by analyzing each step and its impact on the entire equation.- **Problem-Solving Skills:** Methods like squaring both sides to solve \( \sqrt{x+3} \) problems develop necessary problem-solving skills, which are applicable in real-world scenarios.- **Confidence Building:** Successfully solving mathematical problems boosts confidence in one’s ability to tackle difficult tasks.
Mathematics education goes beyond calculations, aiming to foster a deeper understanding and appreciation of how math shapes our world. By practicing regularly and facing diverse challenges, students build a solid mathematical foundation.