Problem 45 - Beginning and Intermediate Algebra

Question

Solve. $$\sqrt{b+3}-2=1$$

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{b+3}-2=1\) is \(b=6\).

Step 1: Isolate the square root term

To isolate the square root term, \(\sqrt{b+3}\), we will add 2 to both sides of the equation: \[\sqrt{b+3}-2+2=1+2\] This simplifies to: \[\sqrt{b+3} = 3.\]

Step 2: Square both sides to eliminate the square root

Now we will eliminate the square root by squaring both sides of the equation. \[\left(\sqrt{b+3}\right)^2=3^2\] This results in: \[b+3=9.\]

Step 3: Solve for b

To solve for \(b\), we will subtract 3 from both sides of the equation: \[b+3-3=9-3.\] This simplifies to: \[b=6.\] The value of \(b\) that satisfies the given equation is \(b=6\).

Key Concepts

Isolating Variables

In algebra, isolating variables is a crucial step when solving equations. The aim here is to "isolate" the variable on one side of the equation to make solving it simpler. For example, consider the equation \( \sqrt{b+3} - 2 = 1 \). Before we can solve this equation, we want the term containing the variable \( b \) to be by itself on one side of the equation.

• Start by looking at the terms around the variable. Identify the operations being performed. In our example, the term \( \sqrt{b+3} \) is reduced by 2.


• To isolate \( \sqrt{b+3} \), we add 2 to both sides of the equation. This cancels out the -2 on the left side: \( \sqrt{b+3} - 2 + 2 = 1 + 2 \).


• The equation simplifies to \( \sqrt{b+3} = 3 \), successfully isolating the square root term containing the variable.

Aligning the goal of having one instance of the variable on one side allows you to focus specifically on its calculation without extra distractions.

Eliminating Square Roots

Square roots can make working with equations a bit tricky as they signal the "opposite" of squaring a number. Eliminating square roots is a method used to simplify equations so that variables are easier to solve for. Once you've isolated the square root, the next logical step is to get rid of it.

• With the isolated square root term \( \sqrt{b+3} = 3 \), we can proceed to eliminate the square root by squaring both sides of the equation: \( (\sqrt{b+3})^2 = 3^2 \).


• This effectively removes the square root. Squaring cancels the square root, turning \( \sqrt{b+3} \) into simply \( b+3 \).


• Now the equation becomes \( b+3=9 \).

Squaring both sides is a tool to simplify the appearance of variables and make solving equations more approachable. It's important to ensure the isolated term is simplified before attempting this to avoid errors.

Step-by-Step Algebra Solutions

Solving algebraic equations can be straightforward when tackled with a clear, step-by-step approach. This method ensures you're addressing each part of the equation methodically, reducing the risk of mistakes. Here's how the problem is approached:

• Identify and isolate the complex term first. In our case, the square root term is separated to make further steps easier.


• Perform operations symmetrically on both sides of the equation. This keeps the equation balanced and accurate throughout the solving process.


• Finally, simplify each side of the equation one step at a time. Work through it incrementally rather than rushing, ensuring every operation is precise.

This methodical approach not only aids in solving the current problem but also trains the mind to deal with different variants of algebraic challenges efficiently.

Simplifying Equations

Simplifying equations involves making them easier to solve by reducing complexities within them. Converting a problem into simpler parts or a simplified form means fewer operations to perform and, often, a clearer path to the solution.

• Start by reducing unnecessary elements. Here, the subtraction of 2 from \( \sqrt{b+3} \) is first "simplified" by restitution, or adding 2 to balance both sides.


• Reduce complex expressions by removing radicals or powers when necessary. In our example, this involved squaring the root, which resulted in a simple linear equation \( b+3=9 \).


• After simplifying, execute basic arithmetic operations to solve for the variable. Once \( b+3=9 \) is achieved, basic subtraction isolates \( b \).

Breaking down an equation to its simplest terms is an invaluable skill. It not only makes the current problem simple but builds confidence for tackling more complex algebraic tasks ahead.