Problem 44 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check for extraneous solutions. $$ \sqrt{x+5}=7 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 44\).

Step 1: Square both sides

To eliminate the square root on the left side of the equation, you need to square both sides of the equation, which gives \((\sqrt{x+5})^2 = 7^2\), simplifying to \(x+5 = 49\).

Step 2: Isolate the variable

The next step is to isolate the variable \(x\). This is done by subtracting 5 from both sides of the equation, resulting in \(x = 44\).

Step 3: Check for extraneous solutions

To ensure the solution isn't extraneous, you need to check it in the original equation. Substituting \(x = 44\) into the equation we get \(\sqrt{44+5} = 7\), which simplifies to \(7 = 7\). This is a true statement, so \(x =44\) is a valid solution.

Key Concepts

Square Roots

Square roots are inverse operations to squaring a number. They help us find a number which, when multiplied by itself, gives the original number. For example, the square root of 49 is 7 because 7 squared equals 49. In equations, square roots are often denoted with the radical symbol \( \sqrt{} \).

A common challenge when solving equations involving square roots is removing the radical to make the equation simpler to solve. This is typically done by squaring both sides of the equation. For example, in the equation \( \sqrt{x+5} = 7 \), squaring results in removing the square root, leading to \( x+5 = 49 \).

It's important to understand that when we square both sides, we must consider all possible roots, which can introduce complexities in solving the equation, such as extraneous solutions.

Extraneous Solutions

Extraneous solutions are solutions that emerge from the algebraic solving process but do not satisfy the original equation. This can occur when operations like squaring both sides of an equation introduce solutions that don't actually fit with the initial problem.

During the process of squaring to eliminate square roots, such as in our exercise \( \sqrt{x+5} = 7 \), it's crucial to verify the solutions. After finding a potential solution, we plug it back into the original equation to confirm its validity. Here, we found that \( x = 44 \) solves the equation since \( \sqrt{44+5} = 7 \), confirming that 44 is a true solution.

Always remember to check for extraneous solutions whenever you manipulate equations, especially when dealing with roots or fractions. This practice ensures that the solution is truly correct.

Isolation of Variables

Isolation of variables means rearranging an equation to get the unknown variable by itself on one side of the equation. This is a core step in solving algebraic equations, allowing easier interpretation and solution of the equation.

For example, to solve \( x + 5 = 49 \), we isolate \( x \) by performing inverse operations, which in this case involves subtracting 5 from both sides. This practice leads us to \( x = 44 \).

Understanding how to isolate variables is crucial because it helps break down complex problems into simpler parts. By systematically moving terms from one side of the equation to the other, we can more easily find the value of variables, ultimately solving equations of varying complexity.