Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ 1+\sqrt{x}=3 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 4 \), and the solution set is \( \{ 4 \} \).

Step 1: Isolate the Square Root Expression

Subtract 1 from both sides of the equation to isolate the square root expression. \[ 1 + \sqrt{x} = 3 \] becomes \[ \sqrt{x} = 3 - 1 \] which simplifies to \[ \sqrt{x} = 2 \].

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square root. \[ (\sqrt{x})^2 = 2^2 \] which simplifies to \[ x = 4 \].

Step 3: Check the Solution

Substitute \( x = 4 \) back into the original equation to ensure the solution is correct. Start with \[ 1 + \sqrt{4} = 3 \]. Since \( \sqrt{4} = 2 \), the equation becomes \[ 1 + 2 = 3 \], which is true. Thus, \( x = 4 \) is correct.

Step 4: Write the Solution Set

Since the equation has only one solution, the solution set is simply \( \{ 4 \} \).

Key Concepts

Understanding Square Roots

A square root asks the question: what number multiplied by itself results in the given number? For example, the square root of 4 is 2 because 2 times 2 equals 4. Hence, when solving equations involving square roots, such as \( \sqrt{x} = 2 \), we are looking for a number \( x \) that when square rooted gives back 2. Square roots are denoted by the symbol \( \sqrt{} \), and they are essential in many mathematical problems, like the one we are working with.

• If you know the result of the square root, you can "solve backward" by squaring to find the original number.
• Square roots can be tricky, especially when negative numbers are involved, as the square root of a negative number results in an imaginary number.

It's important to recognize that squaring a square root effectively cancels it out, leaving you with the number under the square root sign.

Isolation of Variables

Isolating the variable is the process of getting the unknown variable by itself on one side of an equation. This is done by performing operations that simplify the equation. In the given exercise, the goal was to isolate \( \sqrt{x} \). Steps involved include:

• Subtracting 1 from both sides to move all numbers away from \( \sqrt{x} \).
• After this step: you get \( \sqrt{x} = 2 \).

Isolation is crucial because it prepares the equation for the next steps, such as squaring, that bring us closer to finding the actual value of the variable. This systematic approach will allow you to solve equations more effectively and gain confidence in handling both simple and complex problems.

Defining and Finding the Solution Set

The solution set contains all the possible solutions to an equation, and sometimes, there could be more than one solution. In the given problem, after isolating \( x \) and squaring both sides, we found \( x = 4 \). Hence, the solution set is \( \{ 4 \} \).

• The solution set provides a complete answer to the equation.
• It is written in curly braces to denote that these are the values that satisfy the equation.

Understanding the concept of solution sets is important as it ensures you acknowledge all possible answers an equation may present. In cases where there are multiple solutions, all valid results need to be included.

Importance of Checking Solutions

Checking your solutions is a vital step. It verifies that your solution satisfies the original equation. After finding \( x = 4 \), we substitute this value back into the original equation: \( 1 + \sqrt{4} = 3 \). Simplifying this gives \( 1 + 2 = 3 \), confirming that the left-hand side matches the right-hand side.

• Checking solutions ensures you haven't made any miscalculations.
• It builds confidence in your answer being correct.

This practice is especially useful when dealing with more complicated equations. It also reinforces your understanding by retracing the logic of each step and checking that you followed the correct processes.